Random Stability of Random Variables

RANDOM ST ABILITY OF RANDOM V ARIABLES ANDREY SARANTSEV Abstract. F or a random v ariable N = 0 , 1 , 2 , . . . we study the following question: When do es the sum of N many indep enden t and iden tically distributed copies of a random v ariable X hav e the same la w a a non trivial rescaling of X ? W e sho w that suc h N -stable random v ariable exists if and only 1 < E [ N ] < ∞ . Under an additional assumption E [ N ln N ] < ∞ , w e describ e all N -stable X . W e also study a conv erse problem: F or a giv en X ≥ 0 with E [ X ] = 1, w e study the set of all N suc h that X is N -stable. Distributions of N form a semigroup with resp ect to comp osition of probabilit y generating functions. W e show these probabilit y generating functions need to commute with resp ect to c omposition. W e presen t explicit families of comp osition semigroups. Equiv alent formulations hav e app eared in difference forms, and this article aims to unify and extend them. 1. Intr oduction 1.1. The main definition. The topic of this article is the concept called r andom stability or N -stability. Definition 1. T ak e a random v ariable N with v alues 0 , 1 , 2 , . . . with distribution P . A probabilit y measure Q on R , or, equiv alently , a random v ariable X ∼ Q , is called N -stable or, equiv alen tly , P -stable , if there exists a c > 0 suc h that the following equalit y in la w holds: (1) X 1 + . . . + X N d = cX , where X 1 , X 2 , . . . ∼ Q are independent copies of X , indep endent of N . of independent iden tically distributed random v ariables, indep endent of N . Alternativ ely , the article [40] uses the term br anching stability to stress connection with branc hing pro cesses. W e will see in our article that this connection is indeed critical. In [24, Definition 3], the term strictly N -stable is used. Definition 1 is a generalization of a w ell-known concept of strictly stable distributions Q , whic h are N -stable in terms of this Definition 1 for a constant N = n for each n = 2 , 3 , . . . The theory of strictly stable distributions is classic, see [47, 53]. Imp ortan tly , w e can rewrite the main equation from Definition 1. T aking the c haracteristic function (F ourier transform) f ( u ) = E [ e iuX ] of X , and the probability generating function φ of N : φ ( s ) = E [ s N ], w e rewrite this as the Poinc ar e functional e quation : (2) φ ( f ( u )) = f ( cu ) , u ∈ R . If X ≥ 0, then the same equation holds for the Laplace instead of the F ourier transform. Date : March 31, 2026. 2020 Mathematics Subje ct Classific ation. 60E07, 60E10, 60J80. Branc hing pro cesses, stable distribu- tions, strong stability , characteristic function, Linnik distribution, Mittag-Leffler distribution, Y ule pro cess, P oincare functional equation. 2 ANDREY SARANTSEV 1.2. Motiv ation. F or random stability , there has b een a lot of researc h starting from the 1990s, as w ell as some earlier researc h in relation to the classic theory of branc hing pro cesses. But existing literature is fragmen ted. Equiv alent form ulations hav e appeared in difference forms, and this article aims to unify and extend them. It con tains exposition, new research, and critique of existing research. Theorems 1, 2, and 3 are our main results. W e could not find Theorem 1 in other articles. Theorem 2 was shown in [32] but under sp ecial assumptions. Theorem 3 was shown in sp ecial cases, see [13]. 1.3. Organization of the article. Section 2 contains main results: Theorems 1, 2, and 3, as w ell as most lemmas. In Section 3, w e fix a distribution Q on the real line and study G ( Q ), whic h is the set of distributions P on 0 , 1 , 2 , . . . for which Q is P -stable. In Section 4, w e discuss tw o different but equiv alen t definitions of random stabilit y . Section 5 discusses sev eral cases of such semigroups. Section 6 discusses the case of commuting semigroups of probabilit y generating functions equiv alen t to con tinuous-time branc hing processes. The last t wo sections are dev oted to pro ofs and conclusions. 2. Main Resul ts 2.1. Existence results. Theorem 1 is the main result of this article. As men tioned earlier, w e could not find it in the literature, despite its imp ortance. The closest w as [37, Corollary 1.2] with (in their notation) A i = 1 /c . This discusses a generalization of random stabilit y to so-called it smo othing problems. But their pro of applies only to nonnegative N -stable X . Theorem 1. Assume N is not identic al ly zer o or one. Ther e exists an N -stable r andom variable X which is not identic al ly zer o if and only if 1 < E [ N ] < ∞ . The follo wing is a w ell-known uniqueness result, whic h can b e found in the classic mono- graph on branching pro cesses: [1, Chapter I], Section 10, Theorem 2. Prop osition 1. Assume E [ N ] > 1 and E [ N ln N ] < ∞ . Then an N -stable X ≥ 0 is unique in law under c ondition E [ X ] = 1 . Then c = E [ N ] in (1) . An earlier monograph [17, Chapter 1], Theorem 7.3, has this result for the case E [ N 2 ] < ∞ . The following result w as prov ed in [1, Chapter I], Section 10, Theorem 3. Let ( N k ) b e a branc hing pro cess with N offsprings and c = E [ N ] ∈ (1 , ∞ ). This case is called sup er critic al . There exists a sequence ( C k ) of p ositiv e n umbers such that (3) C k +1 C k → c, N k C k a.s. → X, where X ≥ 0 is an N -stable random v ariable. Remark 1. There are tw o cases, when c = E [ N ] ∈ (1 , ∞ ); see also discussion in [5]. F or a conceptual proof, see [39]. F or other proofs, see [49, Theorem 3.1], [50, Theorem 4.3]. • If E [ N ln N ] < ∞ , w e can simply take C k = c k . Then E [ X ] = 1. • If E [ N ln N ] = ∞ , w e need to tak e C k  = c k , and E [ X ] = ∞ . Definition 2. W e call such X from Remark 1 the standar d N -stable r andom variable , and the distribution of X is called the standar d N -stable distribution . RANDOM ST ABILITY OF RANDOM V ARIABLES 3 2.2. Strictly stable distributions on the real line. Recall a well-kno wn classification of stictly stable distributions Q , see [47, Chapter 1] or [53, Chapter V]. Its characteristic function (F ourier transform) Z R e i ux Q (d x ) = e − g ( u ) , i = √ − 1 , can be represented as (4) g ( u ) = ( β + γ isgn( u )) | u | α with α ∈ (0 , 2], β > 0, and the following additional restrictions: • if α = 2 then γ = 0 and Q = N (0 , 2 β ); • if α = 1 then γ is unrestricted and Q is shifted Cauch y; • if α ∈ (0 , 1) ∪ (1 , 2) then | γ | ≤ | tg ( π α/ 2) | . Of course, we also ha ve the trivial case β = γ = 0, when Q = δ 0 . Denote the set of such functions g as G . Suc h distribution Q is also called strictly α -stable to stress its dep endence up on the index α . Note that the function g satisfies (5) g ( cu ) = c α g ( u ) , c > 0 , u ∈ R . A standard reference is the classic monograph [47, Chapter 1]: Equiv alen t Definitions 1.1.1, 1.1.4, 1.1.5, 1.1.6 of stable distributions, and Prop erties 1.2.6, 1.2.8 of strictly stable distri- butions. Also, see other monographs: [38, Chapter 3]: Theorem 5.7.3 and discussions after this; a more recen t b o ok [53, Chapter V]: Section 7 for general results ab out strict stabilit y; Theorem 7.6 for strictly stable symmetric distributions; Theorem 3.5 for p ositiv e strictly stable symmetric distributions; Theorem 3.5 for p ositive strictly stable random v ariables. Finally , we p oin t out to [13, Chapter 4], section 4.5, page 130. W e stress that muc h of classic researc h inv olv es stable not strictly stable distributions, whic h arise as weak limits (6) X 1 + . . . + X n a n − b n , n → ∞ , for indep enden t iden tically distributed X 1 , X 2 , . . . But we use strictly stable distributions here in this article, with b n = 0. 2.3. Classification of N -stable X for giv en N . In Theorem 2, we classify N -stable distributions. This replicates the results [32, Prop osition 2.1, Proposition 3.1] but without an artificial additional assumption in Remark 2 below. The N -stable laws are represented via a product inv olving a standard N -stable random v ariable, and an independent strictly α -stable random v ariable. Theorem 2. Assume E [ N ] > 1 and E [ N ln N ] < ∞ . Any N -stable X c an b e r epr esente d as the pr o duct (7) X d = Y 1 /α Z, wher e Y is the standar d N -stable r andom variable, and Z is a strictly α -stable r andom variable indep endent of Y , for some α ∈ (0 , 2] . The char acteristic function of X c an b e r epr esente d as (8) f ( u ) = E  e i uX  = L ( g ( u )) , 4 ANDREY SARANTSEV wher e L ( u ) = E [ e − uY ] is the L aplac e tr ansform of this standar d N -stable Y , and g ∈ G is such that the char acteristic function of Z is e − g . Mor e over, if X is N -stable with index α , then it satisfies (1) with (9) c = ( E [ N ]) 1 /α . Separately , w e state an imp ortant uniqueness lemma. Lemma 1. Assume L 1 and L 2 ar e L aplac e tr ansforms of two pr ob ability me asur es Q 1 and Q 2 on the p ositive half-line with me an 1 . L et g 1 , g 2 ∈ G b e such that L 1 ◦ g 1 ≡ L 2 ◦ g 2 . Then L 1 ≡ L 2 and g 1 ≡ g 2 . 2.4. Remo v al of artificial assumptions. This result is imp ortan t for the follo wing sub- tle reason. Previously , such represen tation result was pro ved by [32] under an additional condition on the Laplace transform L ( u ) = E [ e − uY ] of this standard N -stable Y . Remark 2. F or an y t wo infinitely divisible characteristic functions e − f and e − g on the real line, L ◦ f ≡ L ◦ g implies f ≡ g . This condition w as first in tro duced in [55] and used for the con verse of the tr ansfer the or em. This theorem is ab out a transfer from w eak limits of sums of indep endent random v ariables to w eak limits of random sums of these v ariables, [13, Chapter 4]: Section 4.1, Theorem 4.1.2, and references therein. Says [55]: The fol lowing c ondition is pr esumably true, but so far no one was able to pr ove or dispr ove it. It turns out from our researc h that w e do not, in fact, need to pro v e this condition! Indeed, from 1973 un til no w, to the b est of our kno wledge, no one could pro v e this in the general form. There were many discussions of this condition in a recen t b ook [13], whic h is an unabridged republication from 1996: [13, Chapter 4], Section 2. T o quote: No one has manage d either to pr ove this or to r efute it yet. There, it w as pro ved in sev eral cases. In [22, Chapter 8], it was mentioned but unpro v ed in Remark 8.8.16. 2.5. Examples of random stable v ariables. Here, w e consider the three cases of N -stable distributions: (a) symmetric; (b) Gaussian; (c) on the half-line. Definition 3. The random v ariable X (or its distribution Q ) is called a symmetric N -stable if it is N -stable and X d = − X . Lemma 2. Assume 1 < E [ N ] and E [ N ln N ] < ∞ . A ny symmetric N -stable X has char- acteristic function f ( u ) = L ( β | u | α ) for some β > 0 and α ∈ (0 , 2] , wher e L is the L aplac e tr ansform of the standar d N -stable r andom variable Y . This X c an b e r epr esente d as X d = Y 1 /α Z, E  e i uZ  = exp ( − β | u | α ) . Her e, Z is indep endent of Y and is symmetric α -stable. Definition 4. Any N -stable random v ariable with index α = 2 is called a Gaussian N -stable r andom variable . W e stress its similarity with classic Gaussian random v ariables. In the literature, some- times a Gaussian N -stable X is defined using (1) with (9) with α = 2. An additional restriction E [ X 2 ] = ∞ is in tro duced in [23, Definition 2]. Confusingly , the opp osite restric- tion E [ X 2 ] < ∞ was introduced in [13, Chapter 4], see Definition 4.6.1; or [24, Definition 2], or [22, Chapter 8], Definition 8.3.2. But we do not need these restrictions in the definition of a Gaussian N -stable random v ariable, as seen in the follo wing lemma. RANDOM ST ABILITY OF RANDOM V ARIABLES 5 Lemma 3. Assume E [ N ] > 1 and E [ N ln N ] < ∞ . A ny Gaussian N -stable X is symmetric: X d = − X . It satisfies E [ X 2 ] < ∞ . T aking Y the standar d N -stable r andom variable, with L aplac e tr ansform L , we c an r epr esent X = √ Y Z , wher e Z ∼ N (0 , σ 2 ) is indep endent of Y . The char acteristic function of X is given by E  e i uX  = L ( σ 2 u 2 / 2) . The final case is a nonnegativ e N -stable random v ariable X . It is easier to study their Laplace transforms than characteristic functions. Lemma 4. If E [ N ] > 1 and E [ N ln N ] < ∞ , any nonne gative N -stable X has L aplac e tr ansform E  e − uX  = L ( β u α ) for the L aplac e tr ansform L of the standar d N -stable r andom variable Y , and for some α ∈ (0 , 1] and β > 0 . This X c an b e r epr esente d as X d = Y 1 /α Z , wher e Z is the stable sub or dinator of index α , indep endent of Y , define d via L aplac e tr ansform (10) E  e − uZ  = e − β u α , u > 0 . Remark 3. A nonnegativ e N -stable X can hav e only index α ≤ 1, not α ∈ (1 , 2]. 2.6. Geometric stabilit y. The most classic example is for the geometric N p for p ∈ (0 , 1), with distribution (11) P ( N p = n ) = p (1 − p ) n − 1 , n = 1 , 2 , . . . ; and with exp ected v alue and the probability generating function (12) E [ s N ] = ps 1 − (1 − p ) s , E [ N p ] = 1 p . Then the standard N p -stable distribution of ev ery p ∈ (0 , 1) is the standard exp onential with mean 1: It has densit y and the Laplace transform (13) p ( x ) = e − x , x ≥ 0; L ( u ) = E [ e − uY ] = 1 1 + u , u ≥ 0; Y ∼ Exp(1) . A Gaussian N p -stable distribution is also well-kno wn: The L aplac e distribution with densit y and the characteristic function p ( x ) = 0 . 5 β e − β | x | , x ∈ R ; f ( u ) = 1 1 + β 2 u 2 . More generally , a symmetric N p -stable distribution is kno wn: This is the Linnik distribution with c haracteristic function f ( u ) = 1 1 + β | u | α . with 0 < α ≤ 2. See also [22, Chapter 9], Subsection 9.4.2.1, Theorem 9.4.6. This distri- bution is absolutely con tinuous on the real line, but its density is explicitly known only for α = 2, when it b ecomes the Laplace distribution. The p ositiv e N p -stable distribution has Laplace transform with β > 0 and α ∈ (0 , 1]: L ( u ) = 1 1 + β u α . This is commonly kno wn as the Mittag-L effler distribution ; see [22, Chapter 9], Subsection 9.4.2.2, Theorem 9.4.9. Also, in [13, Chapter 2], these w ere named Kovalenko distributions , 6 ANDREY SARANTSEV see (3.2) or page 30. This distribution is also absolutely contin uous, but its density is explic- itly known only for α = 1 or α = 0 . 5. In the former case, this distribution is exp onen tial, as in (13). In the latter case, for β = 1, it has density f ( x ) = 1 √ π x − 2 e x √ x Z ∞ √ x e − z 2 d z . These were the earliest and most studied examples. An entire c hapter [22, Chapter 9] is dev oted to these geometric stable distributions. See also a great survey [29] with large bibliograph y . Such geometric stable distributions were studied as far bac k as in 1984 in [25] and also in one early article b y T. Kozub o wski [28]. F urther representation results for the Linnik and the Mittag-Leffler distributions are giv en in [27]. 2.7. The infinitely divisible case. The following example is well kno wn, see [17, Chapter I, Example 8.5; Chapter V, Example 13.2], [40, Section 8], [23, Section 6], Exampples 2 and 4, [7, Example 1]; [44, Example 3], and physical applications in [16, Section 8]. Here X is a Gamma random v ariable with shap e 1 /k and certain rate parameter β > 0. W e remind the readers that it has Laplace transform (14) E [ e − uX ] = (1 + β u ) − 1 /k , u > 0 . and N has probabilit y generating function for some p ∈ (0 , 1): (15) φ ( s ) = p 1 /k s (1 − (1 − p ) s k ) 1 /k . And X is N -stable for an y m > 1. This is related to the classic geometric-exp onen tial pair from (13). Indeed, since the sum of k i.i.d. Gamma from (14) is exp onen tial: X 1 + . . . + X k ∼ Exp(1) . And the probability generating function from (15) corresp onds to the random v ariable k M , where M is negativ e binomial with shape 1 /k , and has probabilit y generating function E [ s M ] =  ps 1 − (1 − p ) s  1 /k The sum of k independent identically distributed copies of this M is, in fact, geometric, it is distributed as N p from (11): M 1 + . . . + M k d = N p . In Lemma 5, we generalize this for general infinitely divisible distributions. W e remind the readers the classic definition. F or bac kground, see [53]. Definition 5. A distribution Q on the real line, or, equiv alently , a random v ariable X , is called infinitely divisible if for eac h k = 1 , 2 , . . . , w e can represent X as a sum of k indep enden t iden tically distributed random v ariables X 1 , . . . , X k : X d = X 1 + . . . + X k . In terms of c haracteristic functions f ( u ), or Laplace transforms L ( u ), or probabilit y gen- erating functions φ ( s ), a necessary and sufficien t condition is that for ev ery k = 2 , 3 , . . . the k th ro ot of this function: f 1 /k ( u ), L 1 /k ( u ), φ 1 /k ( u ) m ust also b e a characteristic function, or a Laplace transform, or a probabilit y generating function of some distribution. RANDOM ST ABILITY OF RANDOM V ARIABLES 7 Lemma 5. T ake infinitely divisible r andom variables N = 0 , 1 , 2 , . . . and X on the r e al line. Pick a k = 2 , 3 , . . . and de c omp ose N = M 1 + . . . + M k , X = Y 1 + . . . + Y k into k indep endent c opies of M and Y , r esp e ctively. Then X is N -stable if and only if Y is k M -stable. 2.8. Comm uting distributions. The uniqueness result in Lemma 1 for the representation from Theorem 2 allows us to pro ve the following key statement. First, we consider the c omp osition op er ation (16) φ ◦ ψ ( s ) ≡ φ ( ψ ( s )) for the tw o probability generating functions φ and ψ , or, equiv alently , the corresp onding distributions. This op eration is clearly associative, so the set of all probabiltiy distributions on { 0 , 1 , 2 , . . . } is a semigroup under this op eration. In terms of the random v ariables M and N with probabilit y generating functions φ and ψ , their composition φ ◦ ψ is also a probabilit y generating function of the random v ariable N 1 + . . . + N M , where N 1 , N 2 , . . . are indep endent (of eac h other and M ) copies of N . The unit element for this comp osition op eration is φ ( s ) = s , corresp onding to the v ariable M = 1. Definition 6. W e sa y that the probabilit y generating functions φ and ψ , or their corre- sp onding distributions, or random v ariables M and N , c ommute if φ ◦ ψ ≡ ψ ◦ φ. The k ey result is b elo w. It w as sho wn in other literature in particular cases, but not in this general form. F or such results for or Gaussian N -stable random v ariables, see [13, Chapter 4], Theorem 4.6.1 and Corollary 4.6.1; [24, Theorem 1], [22, Chapter 8], Theorem 8.3.4 and references therein; [44, Theorem 2.1]. The prop erty of comm utativ e semigroups was also sho wn in [29, Remark 2.4]. This survey [29] is v ery comprehensiv e for 1990s. How ev er, they claim that geometric family is unique among suc h explicit commuting families. W e discuss in Section 2 that this is not true. Theorem 3. Assume E [ M ] < ∞ and E [ N ln N ] < ∞ . If X is N -stable, then M and N c ommute if and only if X is M -stable. In this c ase, E [ M ln M ] < ∞ . The standar d N -stable and the standar d M -stable distributions ar e the same. The index α and the r epr esentation of X fr om The or em 2 for b oth M and N ar e the same. Remark 4. As a corollary of Theorem 3, w e get that M and N comm ute if and only if their standard random stable distributions coincide. Lemma 6. Assume E [ N ln N ] < ∞ . If X and Y ar e N -stable, and X is M -stable, then Y is also M -stable. In this c ase, E [ M ln M ] < ∞ , and E [ N ] > 1 , and E [ M ] > 1 . 3. Composition Semigroups 3.1. In tro duction to the problem. Fix a probabilit y measure Q on the real line. In this section, we giv e an ov erview of kno wn results and add a few new ones on the follo wing topic: Find the set of all distributions P on the nonnegativ e integers such that Q is P -stable. Let N ∼ P and assume E [ N ln N ] < ∞ . Then from Theorem 3 and Lemma 6, it is sufficient to consider Q which is the standard N -stable for any such N . This allows to use the Laplace transform L of Q . Suc h X ∼ Q satisfies X ≥ 0 and E [ X ] = 1. 8 ANDREY SARANTSEV Ho wev er, we stress that not ev ery X ≥ 0 can ha ve nontrivial N suc h that X is N -stable. First, to b e N -stable for at least one N with 1 < E [ N ] < ∞ , the distribution of X m ust b e absolutely con tinuous on (0 , ∞ ); see [1, Chapter I], Section 10, Theorem 4; Secton 12, Corollary 1; also [17, Chapter 1], Theorem 8.3. How ever, this absolute contin uit y is v ery far from a sufficient condition for existence of N suc h that Q is N -stable. Recall the pair of negative binomial N and Gamma X random v ariables from (14) and (15) suc h that X is N -stable. Compare it with Lemma 7. Lemma 7. A Gamma r andom variable with shap e p ar ameter α  = 1 /n for n = 1 , 2 , . . . is not N -stable for any N . The idea of the pro of of Lemma 7 is to consider the b eha vior of the probabilit y generating function of N at s = 0. F rom (2), whic h in volv es the b eha vior of the Laplace transform of X at infinit y . This approach is useful to disprov e that φ is a probability generating function in other cases. Such distribution Q migh t hav e an atom at 0, that is, ha v e Q (0) = P ( X = 0) > 0; and this happens if and only if N has an atom at zero: P ( N = 0) > 0. By the w ay , the case of an atom at zero can b e reduced to the case of no atom at zero. See [1, Chapter I], Section 12; [40, Section 9]; [16, Section 3], Definitions 3.1, 3.2; Theorem 3.2. 3.2. Analytic description. Let us describ e suc h N analytically . Let π = P ( X = 0). The Laplace transform L is a one-to-one strictly decreasing function [0 , ∞ ) → ( π , 1], and w e can find its inv erse L ← (denoted using the arrow to distinguish it from 1 /L ): (17) L ( L ← ( s )) ≡ s, s ∈ ( π , 1] . Recall that the functional equation (2) holds for the Laplace transform L of X as well as for its c haracteristic function f : (18) φ ( L ( u )) = L ( cu ) , u ≥ 0 . Using (17), w e rewrite (18) as Schr o der functional e quation : L ← ( φ ( s )) = cL ← ( s ) , s ∈ ( π , 1] . In complex analysis, Schroder’s equation is well kno wn, see [51] for a bibliography and a surv ey of results. W e can also rewrite (18) using (17) as follo ws: (19) φ ( s ) = L ( cL ← ( s )) , s ∈ ( π , 1] . Equation (19) is the k ey form ula in a seminal article [7] on random stability . It will play an imp ortan t role in this article as w ell. This defines the probabilit y generating function on the in terv al ( π , 1] ⊆ [0 , 1]. Ma yb e this is not the en tire [0 , 1], but ev en if this in terv al is smaller, it is large enough to uniquely determine the function φ . Indeed, this function φ , as any probability generating function, is analytic in the unit disc D := { z ∈ C | | z | < 1 } . Therefore, the v alues of φ on this interv al uniquely determine it: This is a classic result in complex analysis. 3.3. Semigroup definitions and prop erties. T ake any probabilit y measure Q on the half-line with mean 1. W e can define tw o semigroups: • The set G ( Q ), or, equiv alently , G ( X ), of all probabilit y generating functions φ (or, equiv alently , all distributions P on { 0 , 1 , 2 , . . . } ) suc h that Q is P -stable. • The set S ( X ) or S ( Q ) of all c ≥ 1 suc h that the righ t-hand side of (19) is a probability generating function. RANDOM ST ABILITY OF RANDOM V ARIABLES 9 F or a given Q , one of the t wo possibilities exist. Either there is no N except N ≡ 1 suc h that (1) holds. In this case, G ( Q ) contains only the identit y probability generating function φ ( s ) ≡ s ; and S ( Q ) = { 1 } . Or Q is N -stable for some non trivial N . The main formula (19) defines a mapping M : S ( Q ) → G ( Q ) as follo ws: (20) M : c 7→ L ( cL ← ( · )) Lemma 8. The set G ( Q ) is close d under c omp osition and we ak c onver genc e. The set S ( Q ) is close d under multiplic ation and under the usual c onver genc e of r e al numb ers. The mapping M fr om (20) is a bije ction, and it pr eserves this semigr oup op er ation and c onver genc e. If 1 is the limit p oint of S ( Q ) , then S ( Q ) = [1 , ∞ ) . Belo w we state an imp ortan t result ab out finite momen ts for distributions in G ( Q ). Lemma 9. Pick a distribution Q on the r e al line, and assume G ( Q )  = { 1 } is nontrivial. Then let N b e any nontrivial r andom variable with distribution fr om G ( Q ) , and let Y b e the standar d N -stable r andom variable. Then: • E [ N ln N ] < ∞ if and only if E [ Y ] < ∞ ; • E [ N ln a +1 N ] < ∞ for a fixe d a > 0 if and only if E [ Y ln a Y ] < ∞ ; • E [ N k ] < ∞ for a fixe d k = 2 , 3 , . . . if and only if E [ Y k ] < ∞ . As a corollary , if one distribution P ∈ G ( Q ) satisfies E [ N ln a +1 N ] < ∞ for a fixed a ≥ 0, or E [ N k ] < ∞ for a fixed k = 2 , 3 , . . . where N ∼ P , then all other distributions in G ( Q ) satisfy this prop ert y , to o. 3.4. P arameterization. Here, w e use c = E [ N ] ≥ 1 (for the standard X , w e ha ve α = 1), and the semigroup op eration on c is multiplication. This exactly matc hes the notation in [7] with e t instead of c , and t ≥ 0. The corresp onding semigroup op eration here is addition. Ho wev er, in other literature, a different parameterization is used: p ∈ (0 , 1). Historically , N - stable X arose as w eak limits of scaled random sums of random v ariables. Much like classic stable distributions are weak lmits of scaled and shifted sums of a deterministic nut growing n umber of indep endent iden tically distributed random v ariables. Therefore, the notation w as used is ν p instead of N , with (21) pν p d → Z ≥ 0 , p ∈ Ξ ⊆ (0 , 1) , p ↓ 0; where Z is, in fact, the standard ν p -stable random v ariable for any p ; and Ξ is a subset of (0 , 1) with limit point 0. They found it more con venien t to imp ose this limit assumption, and not deriv e it from scratc h. Often, another additional condition was imp osed: E [ ν p ] = 1 /p . Normalizing by conv ergence and exp ectation at the same time. But this might not b e conisten t, and w e a void this notation here. The reason is subtle: Match the discussion in Section 1, Remark 1, with the p -notation, and recall the tw o cases, with E [ N ln N ] < ∞ or E [ N ln N ] = ∞ . In b oth cases, w e do ha ve E [ N k ] = c k . • F or the case E [ N ln N ] < ∞ , or equiv alently , E [ Z ] = 1, w e can let p = c − k and ν p = N k . Then E [ ν p ] = 1 p , pν p d → Z, p ↓ 0 . • If E [ N ln N ] = ∞ , or equiv alen tly , E [ Z ] = ∞ , then either let p = c − k ; then E [ ν p ] = 1 /p but not pν p → Z ; or let p = 1 /C k , then pν p → Z but E [ ν p ]  = 1 /p . W e prefer parameterization p = 1 /C k . 10 ANDREY SARANTSEV In the second case, we cannot hav e b oth exp ectation and conv ergence assumptions for ν p . The notation of ν p is used in the follo wing literature (the list is not exhaustive): [44, Section 2], (1) and (3), b oth exp ectation and conv ergence; [32, Section 1], only conv ergence; [23, page 304], they men tioned only exp ectation but clearly meant con vergence as well, as seen from Theorem 2; [24, Section 2], only exp ectation, but later conv ergence is in tro duced; [32, Section 2] (3) and (4), only con vergence; [13, Chapter 4], (6.32), only con vergence. Sometimes, comm utativity is required or deriv ed in special cases. The real reason b ehind the prev alence of this p -notation is that, historically , random stabilit y study to a large extend (although not exclusiv ely) w as focused on geometric stabilit y (that is, when N = N p is geometric from (11)). In this case, pN p con verges to the standard exp onen tial distribution as p ↓ 0, and E [ N p ] = 1 /p . Approximations of sums of N p man y indep enden t iden tically distributed random v ariables as p ↓ 0 (similar in spirit to the Cen tral Limit Theorem) were extensively studied. A go o d exp osition is in [22, Chapter 9]. Lemma 10. Pick a distribution Q on the half-line. Assume G ( Q ) is nontrivial. Then the r andom variable N c with pr ob ability gener ating function (19) with L b eing the L aplac e tr ansform of Y satisfies (22) N c c d → Y , c → ∞ , c ∈ S ( Q ) , if and only if E [ N c ln N c ] < ∞ . In this c ase, E [ Y ] = 1 . 3.5. Explicit-form semigroups. The geometric distribution has the semigr oup pr op erty: The probabilit y generating function G p of N p satisfies (23) G p ◦ G q = G q ◦ G p = G pq . Un til recen tly , this geometric family (and its extensions, see b elo w) w ere the only explicit families of probability generating functions ( G p , p ∈ (0 , 1]), ha ving finite mean G ′ p (1) < ∞ , and this prop erty (23). Extensions include the mo difications of geometric distributions with atoms at zero. Their probabilit y generating functions are general-form fractional-linear. Another extension includes the infinitely divisible case ab ov e. Then X is Gamma with shap e 1 /k instead of an exp onential. Another example is the Sibuya r andom variable N p with probabilit y generating function G p ( s ) = 1 − (1 − s ) p , in tro duced in [52] with β = γ = 1 in (2); see also a more recen t article [33] with an extension with mass at zero. Suc h family of distributions satisfy (23) as w ell. But they ha v e infinite mean. As follo ws from Theorem 1, do not hav e an y N p -stable distributions, for any p ∈ (0 , 1). Un til recently , these were the only kno wn exact-form comm uting semigroups. Ho wev er, recen tly there were breakthroughs on this topic, with sev eral more explicit comm uting fam- ilies of probability generating functions, and with finite mean greater than 1, including a generalization of the Sibuya distribution. This corresp onds to contin uous-time branching pro cesses N = ( N ( t ) , t ≥ 0) (See Section 6 for more details), with (24) G p ( s ) = E  s N ( t )  , p = e − t . F rom (23), the composition of probabilit y generating functions of N ( t ) and N ( t ′ ) is the probabilit y generating function of N ( t + t ′ ). Comparing (21), (22), (24) w e ha v e the follo wing reparametrization rule: (25) p = 1 c = e − t , c = e t = 1 p . RANDOM ST ABILITY OF RANDOM V ARIABLES 11 4. Equiv alent Definitions of Random St ability There is another definition of random stable v ariables in the literature, one that is based on w eak limits of scaled random sums. Just as stable and strictly stable distributions (in the classic sense) are often defined as w ek a limits of scaled sums of N indep enden t identically distributed random v ariables as N → ∞ , so can b e done for random stable distributions. See [32, Section 2], and [40, Theorems 1, 2, 3]. Instead of N → ∞ , w e need a semigroup of probability distributions on nonnegative in tegers; or, equiv alen tly , a semigroup of probabilit y generating functions. W e migh t as w ell assume this semigroup is commutativ e. Although we think one do es not really need this assumption, the pro of is easy if w e imp ose it. Luc kily , ev en when we start from one N suc h that X is N -stable, suc h semigroup comes naturally: This is the discrete semigroup generated b y N ; or, equiv alen tly , the set of proba- bilit y generating functions of N k for eac h k , where ( N k ) is the discrete-time branc hing pro cess with N offsprings. The result b elow is simple but needs to b e included separately . W e could not find it explicitly written in the literature. Lemma 11. T ake a r andom variable Y ≥ 0 with E [ Y ] = 1 . Assume its semigr oup G ( Y ) is not trivial. L et N c b e the r andom variable with pr ob ability gener ating function as in (19) , with c ∈ S ( Y ) . Then a r andom variable X is N -stable for at le ast one, and ther efor e for al l, N with distribution in G ( Y ) , if and only if ther e exists a se quenc e of indep endent identic al ly distribute d r andom variables U 1 , U 2 , . . . indep endent of al l these N , and a function a : S ( Y ) → [0 , ∞ ) such that (26) U 1 + . . . + U N c a ( c ) d → X , c → ∞ , c ∈ S ( Y ) . If this is true, then for any such se quenc e U 1 , U 2 , . . . the statement (26) is e quivalent to (27) : (27) U 1 + . . . + U [ c ] a ( c ) d → Z, c → ∞ , c ∈ S ( Y ) . In this c ase, we have (7) , with Z strictly α -stable, and Y and Z indep endent. W e see that U j b elong to the strict domain of attraction: Recall our discussion ab o v e ab out defining strictly stable distributions as w eak limits of scaled sums of indep endent iden tically distributed random v ariables. W e stress this is different from classic α -stable domains of attraction, where adding constan ts is allo wed: Recall (6) and the discussion there. Strict domain of stability include scaling sums of indep enden t inden tically distributed random v ariables, but not adding constants. See also discussion in [32, Section 2]. 5. Semigr oup Classifica tion No w w e start the discussion of the structure of the semigroup S ( X ) ⊆ [1 , ∞ ) for a giv en nonnegativ e random v ariable X with E [ X ] = 1. It is time to state the main op en question: Consider a distribution Q on [0 , ∞ ) with mean 1. Find all distributions P on { 0 , 1 , 2 , . . . } suc h that Q is P -stable. Find the semigroup structure of G ( Q ). Is it discrete or contin uous? If it is discrete, is it cyclic, or not finitely generated? Can w e generalize some construction tec hniques for the general case? Below, w e pro vide a partial classification of suc h semigroups. The open problem is to complete this classification. 12 ANDREY SARANTSEV 5.1. T rivial semigroup. S ( X ) = { 1 } is the trivial semigroup. Examples: Lemma 7 ab ov e, or X whic h is not absolutely con tinuous. 5.2. F ull semigroup. S ( X ) = [1 , ∞ ) is the full semigroup. Example: geometric N p from (11), with p ↔ 1 /p as the isomorphism. More general discussion on this case, whic h is equiv alent to contin uous-time branc hing pro cesses, is in Section 3. 5.3. Cyclic semigroup. S ( X ) = { 1 , a, a 2 , . . . } for some a > 1. A construction was giv en in [7, Examples 2, 4] based on [9, Section 5] and [4] but in our view the pro ofs are not witten well, and they should b e impro ved. Particular cases of Q discussed in [4, Section 6] are as follo ws: The standard P -stable Q for the following P : (a) P ( { 1 } ) = P ( { 2 } ) = 0 . 5, with φ ( s ) = ( s + s 2 ) / 2; (b) P is the shifted geometric, with probabilit y generating function φ ( s ) = s 2 / (4 − 3 s ), used in [2, Section 3] as a to ol to construct Bro wnian motion on the Sierpinski gasket. The idea is to create an almost constan t solution θ : (0 , ∞ ) → (0 , ∞ ) to the functional equation θ ( ax ) ≡ bθ ( x ), for fixed a, b > 0. See [57] for additional information, also [48, Section 4]. Also, this semigroup corresp onds to a discrete-time sup ercritical branching pro cess ( N k ) with N offsprings, with E [ N ln N ] < ∞ and E [ N ] = a . W e consider X = lim( N k /a k ). Bey ond the general statement that X must b e absolutely contin uous on (0 , ∞ ), there w as researc h on the b eha vior of X at zero and infinity . There are tw o very distinct cases: Sc hr¨ oder, with P ( N = 0) + P ( N = 1) > 0, and B¨ ottc her, with N ≥ 2 almost surely . The former case w as discussed in a v ery recen t article [35]. Both cases were studied in an early article [5]: Asymptotics at zero for Sc hr¨ oder case in [5, Section 2]; Asymptotics at zero for B¨ ottc her case in [5, Section 3]; Righ t tail was studied in [5, Section 4]. Asymptotics at zero for the B¨ ottc her case w as also studied in [12]. A related question is emb e ddability: Whether the discrete-time branching pro cess ( N k ) can b e represen ted as a skeleton for a con tinuous-time branc hing pro cess N ( k △ t ) for some △ t > 0. This was studied in detail in [21], see also [1, Chapter I I I], Section 12. This is equiv alent to upgrading from S ( Q ) ⊇ { 1 , a, a 2 , . . . } to S ( Q ) = [1 , ∞ ). This is distinct from taking a distribution Q and asking whether S ( Q ) = [1 , ∞ ). T o answ er the latter question, w e refer to Section 6 b elo w. 5.4. Natural n um b ers. S ( X ) = { 1 , 2 , 3 , . . . } . This is true for X = 1, and Q = δ 1 , since L ( u ) = e − u , and L ← ( s ) = − ln s , and for c ≥ 1, w e ha ve: L ( cL ← ( s )) = e c ln s = s c , which is a probabilit y generating function if and only if c = 1 , 2 , 3 , . . . , see [7, Example 3]. 5.5. Complete squares. S ( X ) = { 1 , 4 , 9 , 16 , . . . } . This was discussed in [23, Section 4] for L ( u ) = 1 / c h( √ 2 u ), where ch( x ) = ( e x + e − x ) / 2 is the h yp erb olic cosine, with the inv erse function arcc h( s ) = ln( s + √ s 2 − 1). Lemma 12. The function φ c ( s ) = L ( cL ← ( s )) fr om (19) is a pr ob ability gener ating function if and only if c = n 2 for p ositive inte ger n . W e can view X as the first hitting time of {± 1 } b y the standard Bro wnian motion W = ( W ( t ) , t ≥ 0) starting from zero: X d = inf { t ≥ 0 | | W ( t ) | = 1 } . W e refer to [20, Chapter 2, Section 2.8.C, (8.29)] for x = a/ 2 and a = 2 n . F or c = n 2 , the probability generating function is (28) φ ( s ) = L ( n 2 L ← ( s )) = 1 T n (1 /s ) , RANDOM ST ABILITY OF RANDOM V ARIABLES 13 where T n ( u ) = cos( n arccos( u )) = c h(arcc h( u )) is the n th degree Chebyshev p olynomial. The probabilit y generating function φ in (28) corresp onds to the distribution of the hitting time τ n of {± n } by the simple symmetric random w alk ( S k ), starting from 0: S k = Z 1 + . . . + Z k , P ( Z k = ± 1) = 1 2 , Z k i.i.d. Then τ n = min { k ≥ 0 | | S k | = 1 } . See [11, Chapter XIV], Section 4 for general discussion of this hitting time as the duration of gam bling, and Section 5 for explicit form of the generating function with p = q = 1 / 2, see (5.2) and (5.4) on page 352. W e can in terpret W τ n := inf { t ≥ 0 | | W ( t ) | = n } d = n 2 X . This equalit y in law follo ws from the self-similarity of the Bro wnian motion, see [20, Chapter 2]. W e also refer the reader to [23, Section 3] with an interesting discussion ab out rational probabilit y generating functions. 6. Continuous-Time Branching Pr ocesses 6.1. Construction. Pic k a probability measure Q on [0 , ∞ ) with Laplace transform L . This is the case when S ( Q ) = [1 , ∞ ), or, equiv alently , when for any c ≥ 1 the function (19) is a probability generating function of some distribution on { 0 , 1 , 2 , . . . } . In other words, w e hav e a family of commuting semigroups (23) for p ∈ (0 , 1]; and this corresp onds to a con tinuous-time branching pro cess N = ( N ( t ) , t ≥ 0) via (24). This branc hing pro cess is describ ed as follo ws, as outlined in [17, Chapter 5], Theorem 9.1, and [1, Chapter II I], Sections 2 and 3, see also a short discussion in [53, Chapter V], Section 8. Start, as in discrete time, from N (0) = 1. Fix a distribution H on { 0 , 2 , 3 , . . . } called the gener ating distribution . This H pla ys the role similar to the infinitesimal generator for diffusion pro cesses. Eac h particle which exists at time t is replaced during time in terv al [ t, t + △ t ] with small △ t , with probability △ t with the num b er of particles distributed as H . This contin uous-time branching pro cess contains skeletons: Discrete-time branching pro- cesses, (29) N k = N ( k △ t ) , k = 0 , 1 , 2 , . . . Their offspring distribution has probabilit y generating function G p for p = exp( −△ t ). The con verse question is emb e ddability: Whether a discrete-time branching pro cess ( N k ) is really a sk eleton of some contin uous-time branc hing pro cess ( N ( t ) , t ≥ 0), as in (29), see [21] and [1, Chapter I II], Section 12. A necessary and sufficient condition for existence and non-explosion (lac k of t > 0 suc h that X ( t ) = ∞ ) is, see [17, Chapter 5, (9.3)]: The probabilit y generating function h of H needs to satisfy (30) Z 1 0 . 5 d u u − h ( u ) = ∞ . A particular case is E N ( t ) < ∞ (equiv alently , finite mean c = h ′ (1) of H , whic h implies (30)). In this case, scaling asymptotics holds in contin uous time, [17, Chapter 5, Section 11.3]: (31) e − ( c − 1) t N ( t ) d → Y , t → ∞ . 14 ANDREY SARANTSEV F or suc h Y , for an y c > 1 L ( cL ← ( · )) is a probabilit y generating function; so the ab o v e semigroup is [1 , ∞ ). Using the alternativ e notation from (25) w e ha ve (32) a ( p ) N p d → Y , p ↓ 0 , a ( p ) := p c − 1 . As b efore, Y ≥ 0, and we can express the inv erse Laplace transform L ← of Y as a simple function of h : (33) L ← ( s ) = exp  − Z 1 s h ′ (1) − 1 h ( x ) − x d x  This gives us a wa y to construct man y con tin uous comm uting semigroups ( G p , 0 < p ≤ 1) as in (23) whic h ha ve the common random stable X . See [17, Chapter 5], Section 11, (11.7) or [1, Chapter I I I], Section 8, Theorem 3. 6.2. Y ule and Neveu pro cesses. F or geometric distributions, ( N ( t ) , t ≥ 0) is called the Y ule pr o c ess , in tro duced by Y ule in [56, Section 1] in 1925 for mo deling evolution of the n umber of sp ecies as the pure birth pro cess with birth rate n at state n . See the discussion in [17, Chapter 5, Section 8] or [1, Chapter I I I], Section 5. F or Sibuy a distributions, this is called the Neveu pr o c ess , introduced in an unpublished manuscript [46] by Nev eu in relation to Generalized Random Energy Mo del (GREM) of Derrida. Links to Sherrington-Kirkpatric k mo del and GREM are discussed in [3, Chapter 6]. The exponential Exp(1) from (13) is G p - stable for an y p . Ho wev er, there is no Sibyua-stable random v ariable, since the mean of the Sib yua distribution is zero. Prop erties of the Neveu pro cess are discussed in [18]. They also considered generalizations with Sibuya distribution with an atom at zero in [18]. These branc hing pro cesses hav e nonzero extinction probability . F or the Y ule pro cess with geometric distributions, the generating distribution is P ( H = 2) = 1 , h ( s ) = s 2 . F or the Nev eu pro cess with Sibuy a distribution, the generating distribution is P ( H = n ) = 1 n ( n + 1) , n = 2 , 3 , . . . , h ( s ) = s + (1 − s ) ln(1 − s ) , see also [18, Subsection 1.3, formula (16)]. The result (31) applied to the Y ule pro cess with geometric distributions states e − t N ( t ) d → Y ∼ Exp(1) , t → ∞ ; Using the equiv alent notation as in (25), the result (32) applied to the Y ule pro cess states: p Γ p d → Y ∼ Exp(1) , p ↓ 0 . Here, c = 2, since the mean of the constant random v ariable 2 is clearly 2. F or the Neveu pro cess with infinite mean, a v ery different asymptotics holds: e − t ln N ( t ) → Exp(1) , t → ∞ ; or, equiv alently , after reparametrization as in (25): p ln G p → Exp(1) , p ↓ 0 , See the original pro of in [8] for the discrete time (in their notation, F ( s ) = 1 − (1 − s ) e − t and g ( s ) = s e t so a = 1 and b = e t − 1) and [14, Theorem 3] for the contin uous time ( α = 1 , q = 0 in their notation). RANDOM ST ABILITY OF RANDOM V ARIABLES 15 There exist man y other then Y ule con tinuous-time branc hing pro cesses. Equiv alently , there exist many other than geometric families satisfying (16). This contradicts the previous statemen t [29, Remark 2.4] b y T. Kozub o wski. All these new con tinuous-time branching pro cesses b elow are intermediate b etw een the Y ule pro cess (one particle can only create tw o at a time) and the Neveu pro cess (one particle can create arbitrarily man y , and this random n umber has infinite mean). 6.3. Generating geometric distribution. Here w e present a sp ecial case of geometric generating H . The full treatment is given in a recent article [42], with complicated formulas for mass functions and PGFs, using sp ecial functions. Our example is easily treatable and explicit. W e also refer to another article [43] by the same authors, treating the case of P oisson H . This inv olves complicated special functions, to o. Similar computations w ere done for [41, Theorem 1] b y the same authors; but they consider the sub critical case c < 1, with a.s. extinction; of course, this branching pro cess do es not hav e a limiting X . One p ossible application is to ph ysics: the Bose-Einstein distribution ha ve geometric as one possible limit, see [19, (9)], and this can be connected to the curren t pro cess. T ake the generating distribution P ( H = n ) = 2 1 − n for n = 2 , 3 , . . . Shifted geometric with parameter p = 0 . 5, with PGF h ( s ) = 0 . 5 s 2 (1 − 0 . 5 s ) − 1 . This H generates a contin uous-time branc hing pro cess with finite mean E [ N ( t )] = e 2 t at time t . Its PGF can b e found in exact form: ψ t ( s ) = 2 s s 2 + 4(1 − s ) e 2 t , t ≥ 0 , s ∈ [0 , 1] . The limiting distribution Y from (33) is shifted Mittag-Leffler (or Ko v alenko) with index 1 / 2 and the following Laplace transform: E [ e − uY ] = 2 1 + √ 1 + 4 u , u ≥ 0; There is y et another case of quite explicit form ulas for comp osition semigroups, in the case of geometric H mo dified at zero; and Poisson H . These are related to L amb ert W -function , the solution to the equation W ( x ) e W ( x ) = x . W e will not presen t details here, instead referring readers to recen t literature [41, 42, 43]. They ha ve finite mean and nonzero extinction probabilit y . 6.4. Theta-branc hing pro cesses. In a recent article [36], they in tro duced a classification of 9 cases, see Section 4, with generalization of a Neveu pro cess with Sibuy a distribution. It has four parameters, including θ ∈ [ − 1 , 1]. The case 4 is precisely the Sibuy a distribution mo dified so that there is an atom at zero, and a nonzero extinction probabilit y . F or us, the case 3 is the most in teresting, since it corresp onds to the sup ercritical case, but with finite mean. The probability generating function h of the generating distribution H is, see [36, Section 7]: h ( s ) = s + (1 − s ) 1+ θ − (1 − q ) θ (1 − s ) 1 + θ − (1 − q ) θ ) . The probability generating function of the contin uous-time branc hing pro cess at time t with p = e − t , see [36, Section 4]: G p ( s ) = 1 −  p (1 − s ) − θ + (1 − p )(1 − q ) − θ  − 1 /θ . See [15, Prop osition 7, page 1087] for the distribution of the standard N -stable random v ariable Y from (33). 16 ANDREY SARANTSEV 7. Pr oofs In this section, first we presen t pro ofs of three main results: Theorems 1, 2, 3. Next, we presen t pro ofs of the tw elv e lemmas. Finally , we state and prov e a tec hnical lemma from complex analysis. 7.1. Pro of of Theorem 1. This is the longest proof, and we present a short ov erview. W e split it in 8 steps. Step 1 is introductory and in volv es branc hing pro cesses and random sums of copies of X . Step 2 is E [ N ] ≤ 1. Steps 3–7 are for the case E [ N ] = ∞ . W e symmetrize the distribution X and then use tightness arguments to arrive at a con tradiction. Finally , Step 8 mentions the classic case 1 < E [ N ] < ∞ . Step 1. As b efore, let φ b e the probability generating function of N . Consider a discrete- time branc hing pro cess ( N k ) starting from N 0 = 1 and eac h particle having the num b er of offsprings distributed as N . The probabilit y generating function of N k is the k th composition of φ : φ k ( s ) = E  s N k  = φ ( φ ( . . . φ ( s ) . . . )) . F or background, w e refer to the classic monographs [17, Chapter 1] and [1, Chapter I]. Applying the main equality k times, w e get: (34) X 1 + . . . + X N k d = c k Z 1 . This identit y is helpful for the pro ofs of lack of N -stable X for E [ N ] ≥ 1 and E [ N ] = ∞ . The first case is simple, the second is m uch harder. W e alwa ys assume ( N k ) is indep enden t of X 1 , X 2 , . . . Step 2. Assume E [ N ] ≤ 1. F rom the classic theory of branc hing pro cesses [17, Chapter 1], Theorem 6.1 or [1, Chapter I], Section 5, Theorem 1, the pro cess ( N k ) b ecomes extinct with probabilit y 1. THat is. N k = 0 for some k almost surely . This immediately implies that the left-hand side of (34) is equal to zero. Therefore, the righ t-hand side of (34) is also zero, whic h implies X 1 = 0 almost surely . This con tradiction completes the pro of of lack of non trivial N -stable X if E [ N ] ≤ 1. Step 3. Now assume E [ N ] = ∞ . Define Y n := X n − X ′ n , where X ′ 1 , X ′ 2 , . . . is y et another sequence of copies of X , indep enden t of each other, of X 1 , X 2 , . . . and of the branc hing pro cess ( N k ). Then (35) Y 1 + . . . + Y N k = ( X 1 + . . . + X N k ) −  X ′ 1 + . . . + X ′ N k  . Dividing (35) b y c k , w e get: Σ k := 1 c k ( Y 1 + . . . + Y N k ) = S k − S ′ k , (36) S k := 1 c k ( X 1 + . . . + X N k ) , S ′ k = 1 c k  X ′ 1 + . . . + X ′ N k  . (37) Note that S k and S ′ k are dep enden t via N k . Thus w e c annot claim that Σ k d = X − X ′ for an indep enden t cop y X ′ of X . How ever, w e can claim the sequence (Σ k ) is tigh t (in other w ords, relatively compact, b ounded in probability): F or an y ε > 0, there exists a K > 0 large enough so that P ( | Σ k | ≥ K ) ≤ ε for all k . This follows from the tigh tness of sequences ( S k ) and ( S ′ k ), whic h all ha ve the same distribution, the same as X . Step 4. Define the characteristic function of Σ k : (38) g k ( u ) = E  e i u Σ k  RANDOM ST ABILITY OF RANDOM V ARIABLES 17 It is real-v alued, since Σ k d = − Σ k . It turns out that this simplifies the pro of in a critical wa y . See [38, Chapter 3], Theorem 3.1.2. Applying [54, Chapter 3], Lemma 3.1.3, we deriv e from tigh tness of the sequence (Σ k ) that (39) sup k ≥ 1 | 1 − g k ( u ) | → 0 , u → 0 . Since g k is real-v alued, and by [38, Chapter 2], page 36, g k ( u ) ≤ 1, w e can simply write this statemen t (39) without the absolute v alue. This c haracteristic function g k from (38) can b e represen ted as (40) g k ( u ) = φ k ( g ( u/c k )) , g ( u ) := E  e i uY n  = E h e i u ( X n − X ′ n ) i = | f ( u ) | 2 is the characteristic function of Y n , see [38, Chapter 3], Corollary 2 of Theorem 3.3.1. Step 5. It follo ws from scaling theory for discrete-time branc hing processes, see for example [50, Theorem 4.4] with infinite mean that for a fixed a > 1, the ev en t A = { N k /a k → ∞} has positive probability . F or an y s ∈ (0 , 1), we can rewrite A = n s N k /a k → 0 , k → ∞ o and note 0 ≤ s N k /a k ≤ 1 almost surely . Applying the F atou lemma, w e get: φ k ( s/a k ) = E h s N k /a k i satisfies the upp er limit (41) lim k →∞ φ k ( s/a k ) ≤ E h lim k →∞ s N k /a k i ≤ 0 · P ( A ) + 1 · P ( A c ) < 1 . Step 6. Comparing (41) with (39), w e get: F or any u, s ∈ (0 , 1) and a > 1, there exists a k 0 ( u, s, a ) such that for k ≥ k 0 ( u, s, a ) we hav e: (42) g ( u/c k ) ≥ s/a k . T ake the logarithms in (42) and apply the elemen tary inequality ln y ≤ y − 1 for y > 0: (43) g ( u/c k ) − 1 ≥ a k ln s. Dividing (43) b y a k and letting k → ∞ , w e get: (44) lim k →∞  a − k ( g ( uc − k ) − 1)  ≥ ln s. In the right-hand side of (44), the n um b er s ∈ (0 , 1) is arbitrary . Therefore, w e can take s as close to 1 as w e wish, to make ln s negative but as close to zero as w e wish. Also, recall that b y [38, Chapter 2], page 36, w e ha ve: g ( uc − k ) ≤ 1. Applying all this to (44), we get: (45) lim k →∞ a − k ( g ( uc − k ) − 1) = 0 . Step 7. Rewrite as second-order discrete difference with step t k := 0 . 5 c − k u : 2( g ( uc − k ) − 1) = g ( − uc − k ) + g ( uc − k ) − 2 g (0) = △ t k 2 g (0) where we define △ t 2 g ( u ) := g ( u − 2 t ) + g ( u + 2 t ) − 2 g ( u ). Clearly , t k → 0, and letting a = c 2 (since a is arbitrary), we rewrite (45) as △ t k 2 g (0) /t 2 k → 0. Apply [38, Chapter 2], Theorem 2.3.1 and conclude: g ′′ (0) = 0, thus E [ Y 2 ] = 0, and Y = 0 almost surely , thus g ( u ) ≡ 1. Comparing this with (40), w e get: | f ( u ) | ≡ 1. Apply [38, Chapter 2], Theorem 2.1.4, Corollary 1. W e get: f ( u ) = e i uX 0 and X = x 0 almost surely for some x 0 ∈ R . But 18 ANDREY SARANTSEV X is N -stable. Plugging X = x 0 in to (1), we get: x 0 N = cx . Thus either N = c , whic h con tradicts E [ N ] = ∞ ; or x 0 = 0, which implies X = 0 almost surely , but we exclude this trivial case. This completes the pro of that there is no N -stable X in case E [ N ] = ∞ . Step 8. The classic case 1 < E [ N ] < ∞ is well-kno wn and discussed in the Introduction, using scaling limits of dsicrete-time sup ercritical branching pro cesses. See [1, Chapter I], Section 10, Theorems 2 and 3. 7.2. Pro of of Theorem 2. Step 1. W e sho w that an y random v ariable X with c haracteristic function f = L ◦ g and g ∈ G is N -stable. Apply (9) and (2) with c = E [ N ]: φ ( f ( u )) = φ ( L ( g ( u ))) = L ( cg ( u )) = L ( g ( c 1 /α u )) = f ( c 1 /α u ) . Inciden tally , this prov es that c haracteristic functions of the left- and right-hand sides of (1) coincide. Thus we hav e equalit y in la w. Step 2. W e sho w that an y random v ariable X with characteristic function f ( u ) = E  e i uX  = L ( g ( u )) for some g ∈ G of index α can b e represen ted as X = Y 1 /α Z , where Y is the standard N -stable random v ariable, and E  e i uZ  = e − g ( u ) . W e simply apply [32, Proposition 3.1] and use their remarks on page 309, (10) – (12), Remark 1, ab out the difference b etw een stable and strictly stable. One can also consult [47, Chapter 1], Section 2 for the latter question. Step 3. T ake a characteristic function f of an N -stable random v ariable X . W e need to prov e that this solution to the main equation (2) can b e represented as f = L ◦ g for some g ∈ G . W e apply results [13, Chapter 4], Section 4.6. Recall the discussion ab out parameterization. Match the notation: Θ = { c − 1 , c − 2 , . . . } , ν c − k = N k , ν = X . As in the pro of of Theorem 1, w e conclude that if X is N -stable, then X 1 + . . . + X N k d = c k X 1 . Rewrite this using our new notation: X θ, 1 + . . . + X θ,N k d = X , θ = c − k , X θ,j := X j c k . In the notation of [13, Chapter 4], we ha ve (6.33) with F b eing the CDF of X . Actually , our statemen t is even stronger: W e hav e equality in law, not just con vergence in law. Also, we ha ve (6.32) in the same notation. This was discussed in the subsection on parameterization. Therefore, by [13, Chapter 4], Theorem 4.6.3, the formula (6.34) holds with an infinitely divisible Y with the CDF G and with m ( θ ) = [1 /θ ] = [ c k ]. Rewrite this conclusion in our original notation: X 1 + . . . + X [ c k ] c k d → Y . By the classic theory of stable distributions, this implies Y is strictly stable. And th us E  e i uY  = e − g ( u ) for some g ∈ G . By [13, Chapter 4], Theorem 4.6.5, w e can write (6.30), whic h completes the pro of of the represen tation f = L ◦ g . With this, w e prov ed that a random v ariable X is N -stable if and only if its c haracteristic function f can b e represented as L ◦ g with g ∈ G , and thus completed the proof of Theorem 2. RANDOM ST ABILITY OF RANDOM V ARIABLES 19 7.3. Pro of of Theorem 3. Step 1. F rom Theorem 2, the c haracteristic function f of X can b e represen ted as f ( u ) = L ( g ( u )), where L is the Laplace transform of the standard N -stable Y , and g ∈ G . W e also ha ve (2) with c = E [ N ]. Finally , E [ Y ] = 1. Step 2. Assume M and N comm ute. Then their probability generating functions φ and ψ satisfy (16), and therefore (46) φ ( ψ ( L ( u ))) = ψ ( φ ( L ( u ))) = ψ ( L ( cu )) . The function ψ ◦ L is the Laplace transform of the random v ariable S = Y 1 + . . . + Y M , where Y 1 , Y 2 , . . . are copies of Y independent of eac h other and of M . F rom (46), we see that S is N -stable, and has finite mean E [ S ] = E [ Y ] · E [ M ] = E [ M ] = b . By the uniqueness of the standard N -stable random v ariable, S/b d = Y . If we write this equality in law in terms of Laplace transforms, we hav e: ψ ( L ( u )) = L ( bu ). Therefore, Y is M -stable. Finally , letting α b e the index of g , we get: ψ ( f ( u )) = ψ ( L ( g ( u ))) = L ( bg ( u )) = L ( g ( b 1 /α u )) = f ( b 1 /α u ) . This pro ves X is M -stable as w ell. Step 3. Conv ersely , if X is b oth M -stable and N -stable, then φ ( f ( u )) = f ( au ) and ψ ( f ( u )) = f ( bu ). Applying b oth these iden tities, w e get: ( ψ ◦ φ )( f ( u )) = f ( abu ) = ( φ ◦ ψ )( f ( u )) . If X is a constant, then M and N are also constants, and obviously they comm ute. If X is not a constan t, then f tak es at least one v alue z inside the unit disc D = { z ∈ C | | z | < 1 } , [38, Chapter 2], Theorem 2.1.4, Corollary 2. But f is con tin uous, therefore this v alue z is the limit point of the image of f . Both φ ◦ ψ and ψ ◦ φ are probabilit y generating functions of some distributions. Therefore, they are analytic on D . Applying the classic result from comp elx analysis, w e get (16). Step 4. If X is N -stable and M -stable, then w e hav e: E [ M ] ∈ (1 , ∞ ) and E [ M ln M ] < ∞ ; also E [ N ] ∈ (1 , ∞ ) and E [ N ln N ] < ∞ . Step 5. Finally , the last claim: F or distinct M and N , the represen tation f = L ◦ g has the same L and the same g . This follows from the uniqueness Lemma 1. In particular, the index α for b oth M and N is the same. 7.4. Pro of of Lemma 1. Step 1. It is clear from the represen tation of g j ∈ G in (4): (47) g j ( u ) = | u | α j ( β j + γ j isgn( u )) that Re g j ( u ) > 0 for u  = 0; and g j ( u ) → 0 as u → 0. And b y Lemma 13 (48) 1 − L j ( g j ( u )) g j ( u ) → 1 , j = 1 , 2 , u → 0 . Indeed, L j is the Laplace transform of a probabilit y measure on [0 , ∞ ) with mean 1. Next, (49) L 1 ( g 1 ( u )) = L 2 ( g 2 ( u )) . Comparing (48) with (49), we get: g 1 ( u ) /g 2 ( u ) → 1 as u → 0. Step 2. F rom (47), rewrite for u > 0: (50) g 1 ( u ) g 2 ( u ) = u α 1 − α 2 · β 1 + γ 1 i β 2 + γ 2 i . 20 ANDREY SARANTSEV Since β 1 , β 2 > 0, then D :=     β 1 + γ 1 i β 2 + γ 2 i     > 0 . Therefore, if α 1 > α 2 , then apply the absolute v alue to (50): (51)     g 1 ( u ) g 2 ( u )     = u α 1 − α 2 · D → 0 , u ↓ 0 . This statemen t (51) con tradicts g 1 ( u ) /g 2 ( u ) → 1. Similarly , the same happ ens in α 1 < α 2 . Step 3. Th us α 1 = α 2 , and (50) b ecomes g 1 ( u ) g 2 ( u ) = β 1 + γ 1 i β 2 + γ 2 i . Letting u → 0 again, we see: This num b er must b e equal to one, therefore, β 1 = β 2 , γ 1 = γ 2 , and g 1 ≡ g 2 =: g . T o complete the final step and sho w L 1 ≡ L 2 , recall L 1 ( g 1 ( u )) ≡ L 2 ( g 2 ( u )). The functions L 1 and L 2 are analytic on the half-space H := { z ∈ C | Re( z ) > 0 } , b y Lemma 13. And the set { g ( u ) | u > 0 } has limit p oints in H . By the classic uniqueness result from complex analysis, L 1 ≡ L 2 . This complets the proof of Lemma 1. 7.5. Pro of of Lemma 2. This is another immediate consequence of Theorem 2: X d = − X ⇒ Z d = − Z . The function g from (4) is ev en: g ( u ) = g ( − u ). This happens if and only if γ = 0, so g ( u ) = β | u | α . 7.6. Pro of of Lemma 3. F rom Theorem 2, we get: X = √ Y Z , where Z is strictly 2-stable and therefore Gaussian. Thus E [ X 2 ] = E [ Y ] · E [ Z 2 ] < ∞ , since E [ Y ] = 1, which follo ws from E [ N ln N ] < ∞ . 7.7. Pro of of Lemma 4. Since X ≥ 0, b y assumption, then the strictly stable random v ariable Z with index α must b e nonnegativ e as well. But this could b e true only if α = 1 and Z is a p ositiv e constant; or α ∈ (0 , 1), and Z has Laplace transform (10). This can b e found in [53, Chapter V], Theorem 3.5, or in the classic reference [47]. Then the Laplace transform of X is E [ e − uX ] = E  e − uY 1 /α Z  = EE h e − uY 1 /α Z | Y i = E  e − β u α Y  = L ( β u α ) . 7.8. Pro of of Lemma 5. Using the notation from the b eginning of this article, w e see that the PGF of M is φ 1 /k . Therefore, the PGF of k M is ψ ( s ) = φ 1 /k ( s k ). The Laplace transform of Y is L 1 /k . It is straightforw ard to c heck that (2) holds if and only if ψ ( L 1 /k ( u )) = L 1 /k ( cu ) for u ≥ 0. This completes the pro of. 7.9. Pro of of Lemma 6. This follo ws from tw o consecutiv e applications of Theorem 3. Indeed, if X is N -stable and M -stable, then M and N comm ute. But Y is N -stable, and therefore Y is M -stable. RANDOM ST ABILITY OF RANDOM V ARIABLES 21 7.10. Pro of of Lemma 7. W e express L ( u ) = (1 + u ) − α for the Gamma random v ariable X with shap e α and scale 1. W e find the in v erse function L ← ( s ) = s − 1 /α − 1. F or c > 1, w e ha ve from (19): φ ( u ) = u  u 1 /α (1 − c ) + c  − α . It has T aylor decomposition which con tains u 1+1 /α . This exp onen t is an in teger if and only if α = 1 /n for n = 1 , 2 , 3 , . . . 7.11. Pro of of Lemma 8. Step 1. If 1 is the limit p oin t of S ( Q ), then for ev ery interv al ( a, b ) ⊆ (1 , ∞ ), ho w ev er small, we ha v e a c ∈ S ( P ) ∩ (1 , b/a ). Then for some n , w e hav e c n ∈ ( a, b ). This c n ∈ S ( Q ), which prov es S ( Q ) is dense in [1 , ∞ ): It is intersecting with an y in terv al. Since S ( Q ) is top ologically closed, S ( Q ) = [1 , ∞ ). Step 2. All other claims ab out S ( Q ) and G ( Q ) directly follow from [7, Prop osition 1]. They use the notation e t = c where t ≥ 0 and c ≥ 1, as discussed in (25) but this is equiv alent. 7.12. Pro of of Lemma 9. This was, in fact, already prov en in the literature, since the tails of standard N -stable Y are related to tails of N : It is sho wn in [1, Chapter I], Section 10, Theorem 2, that for an y a ≥ 0, E [ N ln 1+ a ( N )] < ∞ ⇔ E [ Y ln a Y ] < ∞ . Also, it is shown in [6, Theorem 0] that for k = 2 , 3 , . . . , w e hav e: E [ N k ] < ∞ ⇔ E [ Y k ] < ∞ . 7.13. Pro of of Lemma 10. Step 1. Assume E [ Y ] = 1. The Laplace transform of N c /c is (52) E  e − N c u/c  = L  cL ←  e − u/c  . If Y has mean 1, then L ′ (0+) = − 1. Deriv ative of the inv erse function: ( L ← ) ′ (1 − ) = − 1. Also, an elementary calculus result sho ws: e − u/c − 1 ∼ − u c , c → ∞ . Com bining these asymptotics, w e get: (53) cL ←  e − u/c  → ( L ← ) ′ (1 − ) · c ·  − u c  = u. Applying (53) to (52), we get that the left-hand side of (52) conv erges to L ( u ), which is the Laplace transform of Y . This completes the if part. Step 2. Conv ersely , if indeed there is such con vergence, it holds for c = b n as w ell, where b ∈ S ( Q ). But this corresp onds to scaling of a discrete-time branc hing pro cess. Using the aforemen tioned discussion ab out parameterization, we see that suc h scaling w orks only when E [ Y ] < ∞ , which is equiv alent to E [ Y ] = 1. 7.14. Pro of of Lemma 11. Step 1. Let us deriv e (1) from (26). Direct application of [13, Chapter 4], Theorem 4.6.5, with the follo wing notation mathc: θ = 1 /c and m ( θ ) = [ c ], X θ,j = U j /a ( c ), F is the CDF of X , and G is the CDF of Z , and N = Y in (6.32), and (6.30) from [13, Theorem 4.6.3] can b e rewritten as (7). See our discussion of notation ab ov e. Step 2. Con versely , if X is N -stable, then w e can simply tak e U = X . Then w e ha ve equalit y in la w in (26) instead of w eak conv ergence. Step 3. Equiv alence of (26) and (27) also follo ws from [13, Chapter 4], Theorem 4.6.5. 22 ANDREY SARANTSEV 7.15. Pro of of Lemma 12. W e compute L ← ( s ) = (arcch(1 /s )) 2 / 2. Therefore, using (19): φ ( s ) = L ( cL ← ( s )) = 1 c h( √ c · arcc h(1 /s )) . As s ↓ 0, this has asymptotics φ ( s ) = 2 1 − √ c s √ c . This function is analytic if and only if √ c = n is an integer, that is, c ∈ { 1 , 4 , 9 , . . . } . 7.16. Analytical Laplace transform. T ake a random v ariable X ≥ 0. Consider its Laplace transform L ( u ) = E [ e − uX ]. Lemma 13. Assume E [ X ] < ∞ . We c an define L as a c omplex-value d function on the half-plane H = { z ∈ C | Re( z ) > 0 } . This function is analytic on H , and we have: (54) 1 − L ( z ) z → E [ X ] , z → 0 , Re( z ) > 0 . Pr o of. F or z = u + i v ∈ H , w e write | e − z X | = e − uX . Therefore, E | e − z X | = E [ e − uX ] < ∞ , and E [ e − z X ] is w ell defined. It is complex analytic on H . T o this end, w e need to prov e in the complex analytic sense: L ′ ( z ) = E [ − X e − z X ]. But this, in turn, can b e pro v ed as follo ws. T ak e tw o points z , z ′ ∈ H and draw a segmen t b etw een them. It is parametrized as w t = z t + z ′ (1 − t ). W e need to sho w (55) L ( w t ) − L ( z ) = Z [ z ,w t ] E  − X e − uX  d u. This integral o ver the segment is understo o d in the complex analytic sense. This can b e written using real-v alued in tegration as Z t 0 E h − X e − ( z s +(1 − s ) z ′ ) X i ( z ′ − z ) d s. Remo ve exp ectations for a momen t. F rom complex analysis, we get: e − ( z t + z ′ (1 − t )) X − e − z tX = Z t 0  − X e − w s X  ( z ′ − z ) d s. W e need only to sho w that w e need an interc hange of integration and exp ectation. This requires us to use the F ubini theorem. Usually , this theorem is stated for real-v aued functions, but it works equally well for complex-v alued functions. W e need: (56) E Z t 0   − X e − w s X   ( z ′ − z ) d s < ∞ . Assuming Re( z ) ≤ Re( z ′ ) without loss of generality , Re( z ) ≤ Re( w s ) for all s ∈ [0 , 1]. Hence E | − X e − w s X | = E | X e − Re( z ) X | < ∞ . This prov es (56), and with it pro v es (55). Thus L is analytic on H . The prop ert y (54) can b e sho wn similarly , with z = 0, since E [ X e − z X ] = E [ X ] for z = 0. □ RANDOM ST ABILITY OF RANDOM V ARIABLES 23 8. Conclusions and Open Problems This article is an attempt to review and compare v arious streams of existing literature on branc hing pro cesses and random stabilit y . W e added a few results of our own, which we could not find in the literature; therefore this article cannot really b e considered merely a review, but a hybrid b etw een the original research and literature surv ey . A natural direction for further researc h is: F or eac h probability measure Q on [0 , ∞ ) with mean 1, find the semigroup G ( Q ), and describ e the distributions in this semigroup. Another researc h direction is subtle: Extend our results to the case E [ N ] < ∞ but E [ N ln N ] = ∞ . In this article, w e stressed the imp ortance of the condition E [ N ln N ] < ∞ , whic h is not emphasized in some literature. F urther p ossible researc h directions include generalizing the results for man y dimensions. The most obvious generalization is simply to rewrite the main equation (1) with v ector X but scalar N and c . But results are transferred verbatim to this case; see [22, Chapter 10], [26] for multiv ariate geometric stable distributions. Th us we chose not to write a separate section on this topic. Ho wev er, there exists a more substan tial generalization, pioneered by T. Kozub o wski and A. P anorsk a: op er ator stable laws , when the c from (1) is a constan t matrix, not a scalar. See [30] for operator geometric stable la ws, and [31] for general op erator random stable la ws. W e propose to extend our results to op erator random stable distributions. Y et another possible extension is smo othing tr ansformation: F or other random v ariables A 1 , A 2 , . . . indep enden t of eac h other and of X 1 , X 2 , . . . A 1 X 1 + . . . + A N X N d = X . 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