On the computation of classical, boolean and free cumulants

On the computation of classi cal, b o olean and free cum ulan ts E. Di Nardo ∗ , I. Oliv a † August 21, 2018 Abstract This pap er introduces a simple and computationally efficient algo- rithm for conv ers io n formulae b etw een moments a nd cumulan ts. The algorithm provides just one form ula for classica l, bo olean a nd free cu- m ula nt s . This is realized b y using a suita ble p olyno mia l repr esentation of Abel p olyno mials. The algorithm relies on the classica l umbral cal- culus, a symbolic language int r o duced b y Rota and T aylor in [11], that is par ticularly suited to be implemented by using soft ware for symbolic computations. Here w e giv e a MAPL E pr o cedure. Comparisons with ex- isting pr o cedures, esp ecially for c onv ersio ns betw een moments and free cum ula nt s , as w ell as examples of applications to some well-kno wn dis- tributions (classical a nd free) end the pap er. k eyw ords : umb ral calculus, classical cumulant, bo olean cumulant, f ree cumu- lant, Ab el p olynomial AMS subject class ification: 65C60, 05A40, 46L53 1 In tro d uction Among n umber sequences connecte d to random v ariables (r.v.’s), cumulan ts pla y a central role, charact erizing man y r.v.’s found in the p ractice. More- o v er, du e to their p rop erties of add itivit y and in v ariance u nder translation, cum u lants are n ot necessarily connected with momen ts of r.v.’s so that they ∗ Dipartimento di Matematica e Informatica, Universi t ` a degli S t udi della Bas ilicata, Viale dell’A teneo Lucano 10, 85100 P otenza, Italia, elvira.dinardo@unibas.it † imma.oliv a@hotmail.it 1 can b e an alyzed b y u sing an algebraic p oin t of v iew. F or these reasons, cu- m ulants ha ve b een introdu ced not only in the con text of classical probabilit y theory , but also in th e b o olean [14] and the f r ee con text [1 5 ]. Since cumu- lan ts linearize conv olutions of measures - no matter what framew ork they are referr ed to - this pr op ert y allo ws us qu ic k access to test wh ether a giv en probabilit y measur e is a conv olution. The linearit y of classical con v olutions corresp onds to in dep end en t r.v.’s [9 ] and the linea r it y of free conv olutions allo ws u s to recognize fr ee r.v.’s [15]. The b o olean con volution w as con- structed [14] starting exactly from the notion of p artial cumulan ts, this last extensiv ely used in the con text of s to chastic differential equations. In this pap er, w e p rop ose the classical umbral c alculus of Rota and T a ylor [11] as the most natural syntax for co nv ersions b et we en cum ulan ts (c lassical, b o olean and fr ee) and moments, w ithout any sophisticated programming. The b asic devices of the umbral calc u lus are essenti ally t wo. T h e first one is to r epresen t a unital sequence of n umbers by a sym b ol α, cal led an umbra, that is, to represent the sequence 1 , a 1 , a 2 , . . . by means of th e s equ ence 1 , α, α 2 , . . . of p o wers of α via a linear op erator E , r esem bling the exp ectation op erator of r.v.’s. This setting is quite natural in free pr ob ab ility [15]. The second device consists in representing a sequence 1 , a 1 , a 2 , . . . , b y means of distinct umbrae, as it h ap p ens in pr obabilit y theory too f or indep endent and iden tically distr ibuted (i.i.d.) r.v.’s. Thank s to th ese d evices ab o v e all, the um b ral calculus wa s already emplo yed successfully as a syn tax for symb olic computations in statistics [6, 7] by using Maple . Multiv ariate extensions ha ve b een giv en in [5] for cum ulant estima tors a s an a ltern ative to tensor metho ds [2]. The algorithm here prop osed relies on u m b ral parametrizations of cumu- lan ts (classical, b o olean and fr ee) in terms of momen ts and vicev ersa, carried out in [8]. In the free cont ext, suc h a p arametrization in vol ves um b ral Ab el p olynomials. Here w e show th at it is sufficient to change a parameter in the form u la in ord er to get same con versions in cl assical and b o olean theory . So w e trace bac k all the parametrizations to one um bral p olynomial. W e giv e a suitable expan s ion of suc h a p olynomial in order to get a v ery simple algorithm for th e whole matter. The pap er is organized as follo ws. Section 2 is provided for readers una ware of the classical umbral calculus. W e resume termin ology , notation and s ome b asic definitions. Section 3 recalls the um b ral theory of cumulan ts b oth classical, b o olean and free, as w ell as their corresp onding parametriza- tions. In Section 4, w e sh o w that all the parametrizations can b e r eco v ered through o n ly one um bral p olynomial for whic h a fruitful expansion is pro- vided. The MAPLE algorithm for this exp ansion is also given. Comparisons, 2 with pro cedur es a v ailable in the literature (see for example [1]), confirm the comp etitiv eness of the umbral algorithm. The p ap er end s with some exam- ples of h o w to use um br al calculus in order to co m p ute classical, b o olean and free cu m ulants for some classical pr obabilit y laws. 2 The classical um bral calculus In th e follo wing, we r ecall termin ology , notation and some basic definitions of the classical umbral calculus, as in tro duced by Rot a and T a ylor in [11] and sub sequen tly d ev elop ed by Di Nard o and Senato in [3] and [4]. W e also recall those results u seful in dev eloping the um br al alg orith m , giv en in Section 4. W e skip an y pro of: the reader in terested in-depth analysis is referred to [3 ] an d [4]. Classical um br al calculus is a syntax consisting of the follo wing data: a set A = { α, β , . . . } , called the alphab et , whose elements are named umbr ae ; a comm u tativ e in tegral d omain R whose qu otien t field is of c haracteristic zero 1 ; a linear functional E : R [ A ] → R, called evaluation , suc h that E [1] = 1 and E [ α i β j · · · γ k ] = E [ α i ] E [ β j ] · · · E [ γ k ] for an y set of distinct u m b rae in A and for i, j, . . . , k nonn egativ e in tegers (the so-called unc orr elation pr op erty ); an elemen t ǫ ∈ A, called augmentation [10], such that E [ ǫ n ] = δ 0 ,n , for any nonnegativ e in teger n, where δ i,j = 1 if i = j, otherwise being zero; an eleme nt u ∈ A, called unity um bra, suc h that E [ u n ] = 1 , for any nonn egativ e integ er n. A s equence a 0 = 1 , a 1 , a 2 , . . . in R is um br ally represented b y an um br a α when E [ α i ] = a i , for i = 0 , 1 , 2 , . . . . The elements a i are called moments of the um b r a α. This name recalls the device, familiar to statisticia n s, wh en a i represent s the i -t h moment of a r.v. X . Similarly , the factorial moments of an u m b ra α are the elemen ts a (0) = 1 , a ( n ) = E [( α ) n ] for nonnegativ e inte gers n where ( α ) n = α ( α − 1) · · · ( α − n + 1) is the lo w er fact orial. The follo wing um b rae pla y a sp ecial role in the umbral calculus. Singleton umbra. The singleto n umbra χ is th e um b ra suc h that E [ χ 1 ] = 1 and E [ χ n ] = 0 for n = 2 , 3 , . . . . Its factorial momen ts are x ( n ) = ( − 1) n − 1 ( n − 1 In man y applications R is t he field of real or complex numbers. 3 1)!. Bell umbra. The Bell umbra β is the umbra w hose factorial moment s are all equal to 1, i.e. E [( β ) n ] = 1 for n = 0 , 1 , 2 , . . . . Its momen ts are the Bell n u m b ers, that is the n -th co efficien t in the T aylor series expansion o f the function exp( e t − 1) . An um b ral p olynomial is a p olynomial p ∈ R [ A ] . The supp ort of p is the set of all umbrae o ccurring in p. If p and q are t w o um b ral p olynomials, then p and q are unc orr elate d if and only if their supp orts are disjoin t; p and q are umbr al ly e quivalent if and only if E [ p ] = E [ q ] , in symbols p ≃ q . It is p ossible that t wo d istinct um b rae r epresen t th e same sequ ence of mo- men ts, in such case they are calle d similar u mbr ae . More f orm ally t wo um b rae α and γ are similar wh en α n is um br ally equiv alen t to γ n , for all n = 0 , 1 , 2 , . . . in sym b ols α ≡ γ ⇔ α n ≃ γ n n = 0 , 1 , 2 , . . . . Giv en a sequence 1 , a 1 , a 2 , . . . in R there are infin itely man y d istinct, and th us similar umbrae, repr esen ting th e sequence. Thanks to the n otion of similar u mbrae, the alphab et A can b e extended b y inserting the so-called auxiliar y um br ae, resulting from op erations among similar um br ae. This leads to construct a saturated um br al calculus, in whic h auxiliary umbrae are h andled as elemen ts of th e alph ab et. In the follo wing, w e fo cus the atten tion on some sp ecial auxiliary um- brae. W e assume { α ′ , α ′′ , . . . , α ′′′ } is a set of n u ncorrelated um b rae similar to an u m br a α . Dot p o wer. The symbol α . n is an auxiliary um bra denoting the pro d- uct α ′ α ′′ · · · α ′′′ . Moment s of α . n can b e easily reco vered fr om its d efini- tion. Indeed, if the umbra α represents the sequence 1 , a 1 , a 2 , . . . , then E [( α . n ) k ] = a n k for all nonnegativ e in tegers k and n . Dot pro duct. Th e s y mb ol n . α denotes an auxilia r y umbra similar to the sum α ′ + α ′′ + · · · + α ′′′ . So n . α is the u m br al counterpart of a sum of i.i.d. r .v.’s. Momen ts of n . α can b e expr essed using the n otions of in teger partition 2 and dot-p ow er. By using an um bral v ersion of the w ell-kno w n 2 Recall that a partition of an integer i is a sequ ence λ = ( λ 1 , λ 2 , . . . , λ t ) , where λ j are w eakly decreas in g positive integer s suc h that P t j =1 λ j = i. The integers λ j are named p arts of λ. The lenght of λ is the number of its parts and will be indicated b y ν λ . A different notation is λ = (1 r 1 , 2 r 2 , . . . ) , where r j is th e n u mber of parts of λ equal to j and r 1 + r 2 + · · · = ν λ . Note that r j is said to b e the multiplicit y of j . W e use th e classical notation λ ⊢ i to denote “ λ is a p artition of i ”. 4 m ultinomial expansion theorem, we ha ve ( n . α ) i ≃ X λ ⊢ i ( n ) ν λ d λ α λ , (1) where th e su m is o ver all partitions λ = (1 r 1 , 2 r 2 , . . . ) of th e intege r i, ( n ) ν λ = 0 for ν λ > n , d λ = i ! r 1 ! r 2 ! · · · 1 (1!) r 1 (2!) r 2 · · · and α λ = [ α ′ ] . r 1 [( α ′′ ) 2 ] . r 2 · · · . (2) A feature of the classical um br al calculus is the construction of new auxiliary umbrae b y suitable symb olic substitutions. F or example, in n . α replace the inte ger n by an um b r a γ . F rom (1), the new auxiliary umbra γ . α is suc h th at ( γ . α ) i ≃ X λ ⊢ i ( γ ) ν λ d λ α λ , (3) and it is the umbral counterpart of a so-called rand om sum. By usin g (3 ), w e ha ve β . χ ≡ χ . β ≡ u as w ell as ( α . χ ) i ≃ ( α ) i . In th e next section, w e will see that χ . α has a different meaning. Moreo v er w e ha ve ( γ . β . α ) i ≃ X λ ⊢ i γ ν λ d λ α λ . (4) The umbra γ . β . α is called c omp osition umbr a of α and γ and it is suc h that ( γ . β ) . α ≡ γ . ( β . α ) . Th e c omp ositional inverse α < − 1 > of an u m br a α is such that α < − 1 > . β . α ≡ χ ≡ α . β . α < − 1 > . 3 Classical, b o olean and free cu m ulan ts In this sectio n , we r ecall some r esults give n in [8 ] ab out connections b et w een momen ts and cumula nts (classical, b o olean and free). α -cum ulan t umbra. The um bra χ . α, where χ is the singleton um bra, is called α -cumulan t u m br a. Replacing γ by χ and b y virtue of (3), we hav e ( χ . α ) i ≃ X λ ⊢ i x ν λ d λ α λ ≃ X λ ⊢ i ( − 1) ν λ − 1 ( ν λ − 1)! d λ α λ . (5) Since the second equiv alence in (5) recalls the wel l-kn o wn expression of cum u lants in terms of momen ts of a r.v., it is natural to refer to moments of χ . α as cum ulants of α . The α -cumula nt u m br a, usually denoted b y κ α , is deeply studied in [4]. In particular, if κ α is the α -cum ulant um br a, then 5 α ≡ β . κ α . Mo r eov er , b y recalling equiv alence (3) and E [( β ) i ] = 1 f or all nonnegativ e in tegers i, w e ha ve α i ≃ ( β . κ α ) i ≃ X λ ⊢ i ( β ) ν λ d λ ( κ α ) λ ≃ X λ ⊢ i d λ ( κ α ) λ . (6) Theorem 3.1 (Pa r ametrizatio n s) . L et κ α b e the α -cumulant umbr a. F or i = 1 , 2 , . . . we have α i ≃ κ α ( κ α + β . κ α ) i − 1 κ i α ≃ α ( α − 1 . α ) i − 1 . (7) The sym b ol − 1 . α denotes the in verse of an u m br a α , that is the um b ra su ch that α + − 1 . α ≡ ε. α -b o olean cum ulant umbra. Let M ( t ) b e the ordin ary generating fu nc- tion (g.f.) of a r .v. X , th at is M ( t ) = 1 + P i ≥ 1 a i t i where a i = E [ X i ]. W e ha ve M ( t ) = 1 1 − H ( t ) , where H ( t ) = X i ≥ 1 h i t i , and h i are call ed b o olean cumulan ts of X . The umbral theory of b o olean cum u lants has b een introd uced in [8]. In particular, the α -b o olean cumulan t um b ra η α is such that E [ η i α ] = h i . This um br a corresp onds to th e comp o- sition of the um b ra ¯ α, ha vin g momen ts ¯ α i ≃ i ! α i , a n d the comp ositional in verse of ¯ u, having momen ts n ! , in sym b ols ¯ η α ≡ ¯ u < − 1 > . β . ¯ α. By this last equiv alence, w e get ¯ η i α ≃ X λ ⊢ i ( − 1) ν λ − 1 ν λ ! d λ ¯ α λ . The p revious equiv alence allo ws us to express b o olean cum u lan ts in terms of momen ts. In order to get the inv erted exp r essions, w e need of the f ollo w ing equiv alence ¯ α ≡ ¯ u . β . ¯ η α , that h as b een p ro ve d in the Bo olean In version Theorem (cfr. [8]). Again we ha ve ¯ α i ≃ X λ ⊢ i ν λ ! d λ ( ¯ η α ) λ . Observe the analogy b etw een the similarit y ¯ η α ≡ ¯ u < − 1 > . β . ¯ α , and the one c haracterizing the α -cumulan t u m br a κ α ≡ χ . α ≡ u < − 1 > . β . α. Theorem 3.2 (P arametrizations) . L et η α b e the α -b o ole an cumulant umbr a. F or i = 1 , 2 , . . . , we have ¯ α i ≃ ¯ η α ( ¯ η α + 2 . ¯ u . β . ¯ η α ) i − 1 ¯ η i α ≃ ¯ α ( ¯ α − 2 . ¯ α ) i − 1 . (8) 6 α -free cum ula n t um bra. Let us consider a n oncomm utativ e r.v. X , i.e. an elemen t of an unital n oncomm utativ e algebra A . S upp ose φ : A → C is an unital linear functional. The i -th momen t of X is the complex n umber m i = φ ( X i ) while its g.f. is the formal p o wer ser ies M ( t ) = 1 + P i ≥ 1 m i t i . The noncrossing (or free) cum u lan ts of X are the co efficien ts r i of the ordinary p o wer series R ( t ) = 1 + P i ≥ 1 r i t i suc h that M ( t ) = R [ tM ( t )]. Th e u m br al theory of free cumulan ts has b een introd u ced in [8]. As b efore, th e ¯ α -free cum u lant K ¯ α has b een c haracterized so that E [ K i ¯ α ] = i ! r i . I n particular, by using th e L agrange inv ersion form ula [3], the ¯ α -free cum ulan t umbra is such that ( − 1 . K ¯ α ) D ≡ ¯ α < − 1 > D , where α D is the deriv ativ e um b ra of α, i.e. su c h that α n D ≃ n α n − 1 . It is qu ite obvious to observe the difference b et ween the previous equiv alence and the ones c h aracterizing b oth th e α -classical and the α -bo olean cum ulant um br ae. This is w h y the computation of fr ee cum u lants is quite difficult compared with the cla s s ical and b o olean ones. Sp eic h er h as found a wa y to expressing free cum ulants { r n } n ≥ 1 in terms of momen ts { m n } n ≥ 1 (and vicev ersa) by using n on-crossing p artitions of a set [12, 13]. How ever, the resu lting algorithm is quite difficu lt to imp lemen t. Bryc in [1] uses a differen t app roac h. In the next section, w e p rop ose an unifying algorithm, whic h r elies on the f ollo wing parametrizations. Theorem 3.3 (P arametrizations) . L et K ¯ α b e the ¯ α -fr e e cumulant umbr a. F or i = 1 , 2 , . . . we have ¯ α i ≃ K ¯ α ( K ¯ α + i . K ¯ α ) i − 1 K i ¯ α ≃ ¯ α ( ¯ α − i . ¯ α ) i − 1 . (9) The umbral p olynomials x ( x − n . α ) n − 1 are kno wn as umbral Ab el p oly- nomials. 4 The umbral algorit h m The algorithm w e prop ose relies on an efficien t expansion of the follo win g um b ral p olynomial γ ( γ + δ . γ ) i − 1 for i = 1 , 2 , . . . and δ, γ um brae. Prop osition 4.1. If δ, γ ∈ A then γ ( γ + δ . γ ) i − 1 ≃ X µ ⊢ i ( δ ) ν µ − 1 d µ γ µ . (10) Pr o of. By using the binomial exp ansion and equiv alence (1), w e hav e γ ( γ + δ . γ ) i − 1 ≃ i X s =1  i − 1 s − 1  γ s X λ ⊢ i − s ( δ ) ν λ d λ γ λ . (11) 7 Supp ose we consider the partition µ of the in teger i, obtained b y adding the in teger s to the partition λ . Th en w e ha ve γ s γ λ ≡ γ µ and ν λ = ν µ − 1. If c s denotes the multiplicit y of s in λ and m s denotes the multiplicit y of s in µ, then m s = c s + 1 . T h erefore, we h a v e  i − 1 s − 1  d λ = s i i ! (1!) c 1 · · · ( s !) c s +1 · · · c 1 ! · · · c s ! · · · = s m s d µ i , where the last equalit y comes b y m u ltiplying numerato r and d enominator for m s > 0. Recalling that P s m s = i, equiv alence (10) follo ws. In order to ev aluate γ ( γ + δ . γ ) i − 1 via (10), w e need th e f actorial momen ts of δ and the momen ts of γ . Recall that, if w e j ust ha ve information on momen ts δ i , the factorial momen ts can b e reco v ered b y using th e well-kno wn c hange of bases: ( δ ) i ≃ i X k =1 s ( i, k ) δ k , where { s ( i, k ) } are the Stirling num b ers of I kind. In p articular equiv alence (10) allo ws us to giv e any expression of cumulan ts (classical, b o olean, free) in terms of m omen ts and vicev ersa. i) F or classical cum ulants in terms of momen ts, due to the latter of (7 ), we set δ = − 1 . u and γ = α . Here w e fi nd E [( − 1 . u ) i ] = ( − 1) i = ( − 1) i i ! . ii) F or momen ts in terms o f cla ss ical cumulan ts, du e to the first of (7), w e set δ = β and γ = κ α . Here we kn ow E [( β ) i ] = 1 . iii) F or b o olean cumulan ts in terms of moments, due to the latter of (8), w e set δ = − 2 . u and and γ = ¯ α. Here w e fi nd E [( − 2 . u ) i ] = ( − 1) i ( i + 1) ! . iv) F or m omen ts in terms of b o olean cum ulants, due to the first of (8), we set δ = 2 . ¯ u . β and γ = ¯ η α . Here w e ha ve E [(2 . ¯ u . β ) i ] = E [(2 . ¯ u . β . χ ) i ] = E [(2 . ¯ u ) i ] = E [( ¯ u + ¯ u ′ ) i ] = ( i + 1)! . v) F or free cum u lan ts in terms of moments, d ue to the latter of (9), we set δ = − n . u and γ = ¯ α . Here we h a ve E [( − n . u ) i ] = ( − n ) i . vi) Finally , for moments in terms of free cum u lan ts, due to the fir st of (9), w e set δ = n . u and γ = K ¯ α . Here we h a v e E [( n . u ) i ] = ( n ) i . The umbral algorithm in MAPLE is the follo wing: y:=combi nat[’part ition’](i): umbralg := proc(i,f m,y) i! * add(fm[ j] * mul(((g[ x[1]]/x[1 ]!)^x[2])/x[2]!, 8 x=conver t(y[j],mu ltiset)), j=1..nop s(y)); end: In the MAPLE pro cedur e, the factorial momen ts E [( δ ) j − 1 ] are referred by the v ector fm[j] and the momen ts E [ γ k ] are referred b y the v ector g [k] . T able 1 r efers to computational times (in seconds ) r eac hed by using the um b ral alg orithm and the Bryc’s pro cedur e [1], b oth implemen ted in MA PLE , release 7, wh en w e n eed free cumulan ts in terms of moment s 3 . i MAPLE (umbral) MAPLE (Bryc) 15 0.015 0.016 18 0.031 0.062 21 0.078 0.141 24 0.172 0.266 27 0.375 0.703 T able 1: Comparisons of co m putational times n eeded to compu te free cum u - lan ts in terms of moments. T asks p erf ormed on In tel (R) P entium (R),CPU 3.00 GHz, 512 MB RAM. Momen ts of Wigner semicircle distribution. In free pr obabilit y , the Wigner semicircle d istribution is analogous to the Gaussian r .v. in the clas- sical pr obabilit y . Indeed, fr ee cum ulants of degree higher than 2 of the Wigner semicircle r.v. are zero. Th e first column in T able 2 shows mo- men ts of the Wigner semicircle r .v. X , compu ted by the um b ral algorithm. They are compared with Cata lan n umb ers C i (second column), sin ce it is w ell-kno w n that E [ X 2 i ] = C i and E [ X 2 i +1 ] = 0. By using equiv alence (4) it is straight f orw ard to p ro ve that the umbra corresp onding to the Wigner semicircle distribution is ¯ ς . β . ¯ δ , where ς is the u m b ra whose moments are the Catalan num b ers and δ is an umbra ha ving only second momen t equal to 1 , the others b eing zero. In the next section, we will us e again this um b ra in describing the u m b ral syn tax of a Gaussian r .v. Momen ts of Marchenk o-Pastur dis t ribution. In fr ee probabilit y , the Marc henko -Pastur distribu tion is analog ou s to the P oisson r.v. in the clas- sical pr obabilit y . Ind eed, the free cumulan ts are al l equal to a parameter λ . The last col u mn in T able 2 sho ws moments of the Marchenk o-P astur distribution computed b y the um br al algorithm. 3 The outpu t is in the same form of the one given by Bryc’s p rocedu re. 9 i Wigner Catalan Marc henko-P astur r .v. r.v. n u m b ers 2 1 2 λ 2 + λ 3 0 5 λ 3 + 3 λ 2 + λ 4 2 14 λ 4 + 6 λ 3 + 6 λ 2 + λ 6 5 132 λ 6 + 15 λ 5 + 50 λ 4 + 50 λ 3 + 15 λ 2 + λ 8 14 1430 λ 8 + 28 λ 7 + 196 λ 6 + 490 λ 5 + 490 λ 4 + 196 λ 3 + 28 λ 2 + λ T able 2: Momen ts of some sp ecial fr ee distr ibutions. 5 Computing cum ulan ts of some kno wn la ws through the um bral algorithm An um b r a lo oks like the framewo r k of a r .v. with no reference to an y p rob- abilit y space, ju st looking at momen ts. The w a y to recognize the um bra corresp ondin g to a r .v. is to charact erize the sequence of momen ts { a n } . When the sequence exists, this can b e d one by comparing the m oment gen- erating function (m.f.g.) of a r.v. with the so-called generating function (g.f.) of an um b ra. The g.f. of an umbra has b een defined [4] as the f ollo w- ing formal p o we r series f ( α, t ) = 1 + X n ≥ 1 a n t n n ! . In the classical umbral calculus, the con v er gence of the formal p o wer series f ( α, t ) is not relev an t. This means that w e can define the um bra whose momen ts are the same as the moments of a lognormal r.v., ev en if this r.v. do es not admit a m.g.f. So the umbral algorithm allo w s u s to compute classical 4 , b o olean or free cumulan ts of an y r.v. h aving sequence of momen ts { m n } . Once the um b ra co r resp ondin g to the r.v. has b een c haracterized, w e c ho ose δ = − 1 . u, − 2 . u, − n . u in equ iv alence (10), dep ending on wh ether w e need classical, b o olean or free cumulan ts. In the follo wing, we tak e u p again some of the examp les giv en in [1], sho win g how they can b e reco vered through the u m b ral al gorithm by a suit- able c haracterization of the in vo lved umbrae. P oisson r.v. A Poisson r.v. of parameter λ is umbrally r epresen ted b y 4 Recall th at, d ue to their prop erties of additivity and inv ariance under translation, the cumulan ts are not necessaril y connected with the moments of probabilit y distributions. W e can define cumulants of an y r.v. disregarding the question whether its m.g.f. converge s [9]. 10 the um bra λ . β , b ecause f ( λ . β , t ) = exp[ λ (exp( t ) − 1)], whic h is the m.g.f. of a P oisson r.v. The momen ts are [3] a i = E [( λ.β ) i ] = P i k =1 S ( i, k ) λ k , where S ( i, k ) a r e the Stirlin g n u mb ers of second t yp e. So c u m u lan ts of a P oisson r.v. can b e c omp uted via th e u m b ral a lgorithm, taking as inp ut the sequence of momen ts E [( λ.β ) i ]. If the inpu t is the sequence of factorial momen ts { ( − 1) i i ! } , we get classical cumulan ts; if the input is the sequence of factorial momen ts { ( − 1) i ( i + 1)! } , we get b o olean cumulan ts; if the in put is the sequ ence of factorial moment s { ( − n ) i } we get free cum u lan ts (T ables 3 and 4 in [1]). Comp ound P oisson r.v. A comp ound P oisson r .v. S N = X 1 + · · · + X N , where { X i } are i.i.d. r.v.’s and N is a Po isson r.v. of parame- ter λ, is um b rally represente d by the um br a λ . β . α , b ecause f ( λ . β . α, t ) = exp { λ [ f ( α, t ) − 1] } , wh ic h is the m.g.f. of a comp ound P oisson r.v. The for- mal p o wer series f ( α, t ) corresp onds to th e m.g.f. of X i . D u e to equiv alence (4), the momen ts are a i = E [( λ . β . α ) i ] = P λ ⊢ i λ k d λ α λ . Exp onen t ial r.v. An exp onen tial r.v. is umbrally represented b y the um- bra ¯ u λ b ecause f ( ¯ u/λ, t ) = (1 − t λ ) − 1 , w hic h is the m.g.f. of an exp onen tial r.v. with parameter λ > 0 . So its moments are a i = E [( ¯ u λ ) n ] = n ! λ . I n ord er to obtai n th e second column of T able 3 in [1], c ho ose in (10) λ = 1 and { ( − n ) i } as factorial momen ts. Uniform r.v. An uniform r.v. on the interv al [ a, b ] has m.g.f. M ( t ) = e t b − e t a t ( b − a ) = e t a  e s − 1 s  s = t ( b − a ) . The um b ra, with g.f. e s − 1 s , is the in v ers e o f the Bernoulli umbra [11], i.e. − 1 . ι. So an uniform r.v. on the interv al [ a, b ] is umbrally r epresen ted by the um b ra a + ( b − a )( − 1 . ι ). Recall in g th at E [( − 1 . ι ) i ] = 1 i +1 , we get a i = E  [ a + ( b − a )( − 1 .ι )] i  = i X j =0  i j  a i − j ( b − a ) j j + 1 . In order to obtain the last column of T able 3 in [1], choose in (10) a = − 1 , b = 1 an d { ( − n ) i } as factorial momen ts. Bernoulli r.v. A Bernoulli r.v. of parameter p ∈ (0 , 1) has m.g.f. M ( t ) = q + pe t = 1 + p ( e t − 1) . T he u m b ra with g.f. M ( t ) is χ . p . β (see [4] for more details), whose moments are a i = p . Binomial r.v. A binomial r.v. of paramete r s n, a p ositiv e in teger, and p ∈ (0 , 1) is a sum of n i.i.d. Bernou lli r.v.’s. Similarly a binomial r.v. is um b rally represente d b y n . ( χ . p . β ). Due to equiv alence (1), w e ha ve a i = 11 E { [ n. ( χ . p . β )] i } = P λ ⊢ i ( n ) ν λ d λ p ν λ . In order to obtain the second column of T able 4 in [1], choose { ( − n ) i } as factorial moments in (10). Gaussian r.v. A gaussian r.v. with real parameter µ and σ > 0 has m.g.f. M ( t ) = exp  µt + σ 2 t 2 2  . This p o wer series is the g.f. of the umbra µ + β . ( σ δ ), where δ is an um br a such that E [ δ 2 ] = 1 wh ereas E [ δ i ] = 0 for p ositiv e intege r s i 6 = 2 . The follo w ing p rop osition giv es the expression of the n -th moment of the um b r a represent in g the Gaussian r.v. Prop osition 5.1. F or n=1,2,. . . we have a n = E  ( µ + β . ( σ δ )) n  = ⌊ n/ 2 ⌋ X k =0  σ 2 2  k ( n ) 2 k k ! µ n − 2 k . (12) Pr o of. By using the binomial exp ansion, we hav e  µ + β . ( σ δ )  n ≃ n X j =0  n j  µ n − j  β . ( σ δ )  j ≃ n X j =0  n j  µ n − j X λ ⊢ j d λ ( σ δ ) λ . (13) Supp ose λ = (1 r 1 , 2 r 2 , . . . ) , then ( σδ ) λ ≡ ( σ δ ′ ) .r 1 [ σ 2 ( δ ′′ ) 2 ] .r 2 · · · ≡ σ j δ λ . On the o th er hand, d ue to the d efinition of the umbra δ , we hav e E [ δ λ ] 6 = 0 iff the partition λ is of typ e (2 r 2 ). Th us , if j is od d, ther e does not exist an y p artition of this t yp e an d s o  β . ( σ δ )  j ≃ 0. Instead, if j is ev en, sa y j = 2 k , there exists a uniqu e partition of t yp e (2 r 2 ) whic h corresp onds to r 2 = k . F or this partition, we h a ve d λ = (2 k )! (2!) k k ! so that  β . ( σ δ )  2 k ≃ (2 k )! σ 2 k (2!) k k ! . Replacing this last equiv alence in (13), we hav e  µ + β . ( σ δ )  n ≃ ⌊ n/ 2 ⌋ X k =0 n ! (2 k !)( n − 2 k )! µ n − 2 k (2 k !) σ 2 k (2!) k k ! b y whic h the result follo ws. W e hav e prov ed b y um bral tools that momen ts of a Gaussian r.v. can b e expressed b y using a p articular s equence of orth ogonal p olynomials, the Hermite p olynomials : H ( ν ) n ( x ) ≃ ⌊ n/ 2 ⌋ X k =0  − ν 2  k ( n ) 2 k k ! x n − 2 k , with x = µ and ν = − σ 2 . All th e prop erties of Hermite p olynomials can b e easily reco vered by using the Gaussian um bra. W e skip the details. As b efore, the first column of T able 3 in [1] can b e r eco vered from the um b ral algorithm b y using (12) w ith µ = 0 and σ = 1 and c h o osing { ( − n ) i } as factorial moments in (10). 12 6 Ac kno wledgmen ts The authors thank Giusepp e Guarino for his con tribu tion in the implemen- tation of the u m b r al algorithm. References [1] Bryc, W. (2007) Computing moments of free a d ditiv e con vo lu tion of measures. App l. Math. Comp. 194 , 561–567. [2] McCullagh P . (1987) T ensor metho ds in statistics. Monographs on Statistics and Ap plied Probabilit y . Chapman & Hall, Lond on. [3] Di Nardo, E., Senato, D. (200 1) Um bral nature of the Po iss on ran- dom v ariables. Algebraic combinatorics and computer scie n ce, 245 –266, Springer Italia, Milan. [4] Di Nardo, E., Senato, D. (2006) An umbral setting for cumulan ts and factorial moments. Europ ean J. Combin. 27 , no. 3, 394–413 . [5] Di Na r do E., G. Guarino, D. 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