The origin of infinitely divisible distributions: from de Finettis problem to Levy-Khintchine formula

FRA CALMO PRE-PRI NT: www.fracalmo.org Mathematical Metho ds in Economics and Finan ce: V ol. 1 (2006) 37-55 Universit ` a Ca ’ F oscari di Venezia - Dip ar timento di Ma tema tica Applica t a ISSN: 197 1-6419 (Print Ed ition). ISSN: 197 1-3878 (Electr onic Edition) URL: h t tp://www.dma.univ e.it /mmef/ The origin of infinitely divi sible distributions: from de Finetti’s problem to L ´ evy-Khin tc hine form ula F ra ncesco MAINARDI (1) and Sergei R OGOSIN (2) (1) Department of P hysics, Univ ersity of Bologna , and INFN, Via Ir nerio 46 , I-4012 6 Bologna, Italy Corresp o nding Author. E-mail: franc esco.mainard i@unibo.it (2) Department of Mathematics and Mechanics, Bela rusian State University , Nezavisimosti Av e 4 , BY-220 030, Minsk, Bela rus E-mail: rogosin @bsu.by Abstract The article pro vides an historical su rv ey of the early cont ributions on infin itely divisible d istributions starting from the pioneering works of de Finetti in 1929 up to the canonical forms deve lop ed in the thirties b y Kolmogoro v, L´ evy and K hint c hine. P articular atten tion is paid to single out the p ersonal contributions of the ab o ve authors that were published in Italian, F rench or Russian du ring the p eriod 1929- 1938. In App endix w e rep ort the translation from th e Rus sian into English of a f undamenta l pap er b y Khintc h ine pub lished in Mosco w in 1937. Keyw ords. C haracteristic function, infin itely divisible distributions, stoc hastic p ro cesses with ind ep endent incremen ts, de Finetti, Kol- mogoro v , L´ evy , Khintc hine, Gnedenko . M.S.C. classification. 60E07, 60E10, 60G51, 01A70. J.E.L. classification. C10, C16. 1 1 In tro du c tion The purp ose of this pap er is to illustrate ho w the concep t of an infinitely divisible distribution has b een dev elop ed up to obtain the canonical form of its c haracteristic function. Usually historical asp ects on this dev elopmen t are know n thanks to some notes av ailable in t he classical textb o oks b y L´ evy [75] (published in F renc h in 1937 and 1954), by G nedenk o- Kolmogorov [45] (published in Russian in 1949 and tr anslated in to English in 1954) and by F eller [38] (published in English in 1 966 a nd 1971). Similar historical not es can b e extracted from the recen t treatises b y Sato [105] and by Steutel and v an Har n [112]. In our opinion, how ev er, a b etter historical analysis can b e accomplished if one examines the original w orks of the pioneers, namely Bruno de Finetti (1906-19 85) [26 , 27, 28, 29, 3 1], Andrei Nik olaevic h Kolmogorov (190 3-1987) [68, 69], who published in Italia n in R endic onti del la R. A c c ademia Nazionale dei Linc ei , P aul L ´ evy (1886-1 9 71) [7 3 , 74], who published in F renc h in A n nali del la R. Scuola Normale di Pisa , a nd finally Alexander Y a k o vlevic h Khint- c hine 1 (1894-19 59) [62], who pu blished in Rus sian in t he Bul letin of the Mosc ow State University . Notew orth y is the 1938 b o o k by Khin tc hine him- self [65], in Russian, on Limit Distributions for Sums of Indep endent R andom V ariable s . F or the reader interes ted in the bio g raphical notes and bibliogra- ph y of the men tioned scien tists w e refer: for de Finetti to [20, 23, 24, 2 5], for Kolmogorov to [108 , 1 0 9], for L ´ evy to [76, 77] and for Khintc hine to [44]. In spite of the fact that de Finetti w as the pio neer of the infinitely di- visible distributions in view of his 1929-193 1 pap ers, a s is w ell recognized in the literature, the attribute infinitely divis i b le , as noted by Khintc hine in his 1938 b o ok [6 5], first app eared in the Moscow mathematical sc ho ol, precisely in the 1936 unpublished thesis by G.M. Ba wly (1908-194 1) 2 . According to Khin tc hine [65] the name of in fi nitely divisible distributions (in a prin ted ver- sion) is found in t he 193 6 article b y G.M. Baw ly [5], that w as recommende d for publication in the v ery imp ortant starting v olume of the new series of Matemati ˇ ceski Sb ornik. Ho we v er, w e note that this term was not “stably” 1 There is also the translitera tion Khinchin. 2 Gregor y Minkelevich Bawly gradua ted at the Moscow State Universit y in 1 930, de- fended his PhD thesis under g uidance of A.N. Kolmogor ov in 1 936. His scien tific a dvisor had greatly esteemed his results on the limit distr ibutio ns for sums of independent ran- dom v ariables and cited him in his bo o k with Gnedenko [45]. G.M. Bawly lost his life in Moscow in Nov em ber 1941 at a b ombing attack. 2 applied in the article. Tw o alternativ e (and equiv alen t) terms w ere used, namely infinitely = unbesc hr¨ ankt (German) and un b oundedly = un b egr enzt (German), see [5, p. 918 ]. The first for mal definition of an infinitely divisible distribution was g iv en b y Khin tc hine himself [63]. It reads: a distribution of a r andom variable which for any p o s i tive inte ge r n c a n b e r epr esente d as a sum of n ide nti- c al ly distribute d ind e p enden t r ando m v ariables is c al le d an infinitely divisible distribution . W e note that infinitely divisible distributions (already under this name) w ere fo rmerly studied systematically in the 1937 b o ok by L´ evy [75], a nd so on later in t he 1938 b oo k b y Khintc hine [65]. W e a lso note that L ´ evy himself, in his late biographical 19 7 0 b o o klet [76, p. 103], attributes to Khintc hine the name ind´ efiniment divisible . The canonical form of infinitely divisible distributions is kno wn in the literature as L ´ evy-Khintchine formula , surely b ecause it w as so named by Gnedenk o and Kolmogorov [45] in their classical treatise on Limit D i s tributions for Sums of Indep endent R andom V ariables 3 that has app eared in Russian in 1 9 49 and in English in 1954. The plan o f the presen t pap er is as f o llo ws. In Section 2 w e pro vide a surv ey of the kno wn results on infinitely divisible distributions. Then w e pass to presen t the tale on the origin o f these results by recalling, in a historical p ersp ectiv e, the early publications of our four actors: de Finetti, Kolmogorov, L ´ evy and Khin tc hine. Section 3 is dev oted to de Finetti and Kolmogorov, na mely to the so-called de Finetti’s problem (as it w as referred to by Kolmogorov ). Section 4 is dev oted to L ´ evy and Khin tc hine, namely to the origin of the so-called L ´ evy-Khintc hine f o rm ula. T o our kno wledge the original con tributions b y Khin tc hine ha v e nev er b een translated in to English, so w e find it con v enient to rep ort in App endix the English translation of his 19 37 pap er, that has led to the L´ evy-Khin tc hine form ula. W e plan to publish the English translation of the 19 3 8 b o ok by Khin tc hine on Limit D istributions for Sums of Indep e ndent R andom V a ri - ables along with a few related articles of him ( originally in Italian, German and Russian). Concerning our bibliogra ph y , the ma in text and the fo otnotes g iv e refer- ences to some classical publications. How eve r, w e take this o ccasion to edit 3 W e note that the Russian titles o f b oth b o oks by Khintc hine and Gnedenko & Kol- mogorov are ident ical, altho ugh in the r eference lis t (in Russian and in English) of the bo ok by Gnedenko & Kolmo gorov the title of Khintc hine’s prev ious bo o k is in some w ay different ( Limit The or ems for Sums of Indep endent R ando m V ariables ). 3 a more extended bibliograph y on infinite divisible distributions and related topics, that, eve n if non-exhaustiv e, could b e of some intere st. 2 A s urv ey on infinit e ly divisible distribu tions Hereafter w e recall the classical r esults on infinitely divisible distributions just to in tro duce our notations. W e presume that the reader has a go o d kno wledge in the Probability Theory . In the b elow formulations we essen tially follow the treatmen ts b y F eller [3 8 ] and by Luk acs [80]; in the references, ho w ev er, w e ha v e cited sev eral treatises con taining excellen t c hapters on infinite divisible distributions. A pro babilit y distribution F is infinitely divisible iff for eac h n ∈ I N it can b e represen ted as the distribution of the sum S n = X 1 ,n + X 2 ,n + . . . + X n,n (2 . 1) of n indep enden t ra ndom v ariables with a common distribution F n . It is common to lo cate the random v ar ia bles in a n infinite triangular ar r a y X 1 , 1 X 2 , 1 , X 2 , 2 X 3 , 1 , X 3 , 2 , X 3 , 3 . . . X n, 1 , X n, 2 , X n, 3 , . . . , X n,n . . . . . . (2 . 2) whose rows con tain independen t iden tically distributed ( iid ) random v ari- ables. This definition is v alid in any n um ber of dimensions, but for the presen t w e shall limit our atten tion to one-dimensional distributions. It should b e noted that infinite divisibility is a prop erty of the typ e , that is, together with F all distributions differing from F only b y lo cation parameters are infinitely divisible. Stable distributions (henceforth the Gaussian and the Cauc h y distributions) are infinitely divisible and distinguished by the fact that F n differs from F only by lo cation parameters. On accoun t of the conv olutio n pro p ert y of the distribution functions of indep enden t random v ariables, the distribution function F turns out to b e 4 the n -fold con v olution of some distribution function F n ; then, the not io n of infinite divisibilit y can b e in tro duced b y means of the char a c teristic function : ϕ ( t ) := I E { e itX } := Z + ∞ −∞ e itx dF ( x ) . (2 . 3) In fact, for an infinitely divisible distribution its c haracteristic function ϕ ( t ) turns out to b e, for ev ery p ositive integer n , the n -th p o w er of some c harac- teristic function. This means that there exists, fo r ev ery p o sitive in teger n , a c haracteristic function ϕ n ( t ) suc h that ϕ ( t ) = [ ϕ n ( t )] n . (2 . 4) The function ϕ n ( t ) is uniquely determined b y ϕ ( t ), ϕ n ( t ) = [ ϕ ( t )] 1 /n , pro- vided that one selects the principal bra nc h for the n -th ro ot. Since Eqs. (2.2) and (2.4) are equiv alent, alternativ ely one could sp eak ab out infinitely divisible distributions or infinitely divisible ch aracteristic functions. Eleme n tary prop erties of infinitely divisible c haracteristic func- tions are listed b y Luk acs [80]. The concept of infinite divisibilit y is v ery imp ortant in pro babilit y theory , particularly in the study of limit theorems. Here w e stress t he fact tha t infinitely divisible distributions a re in timately connected with sto chastic pr o c esses with ind e p enden t incr ements . By this w e mean a family of random v ariables X ( λ ) dep ending on the con tin uous parameter λ and suc h that the incremen ts X ( λ k +1 ) − X ( λ k ) are m utually indep enden t for any finite set { λ 1 < λ 2 < . . . < λ n } . More precisely the pro cesses are assumed to b e homo gene ous , that is with stationary incr e m ents . Then the distribution of Y ( λ ) := X ( λ 0 + λ ) − X ( λ 0 ) dep ends only on the length λ of the in terv al but not on λ 0 . Let us mak e a partition the interv a l [ λ 0 , λ 0 + λ ] b y n + 1 equidistan t p o ints λ 0 < λ 1 < . . . < λ n = λ 0 + λ and put X k ,n = X ( λ k ) − X ( λ k − 1 ). The n the v ariable Y ( λ ) of a pro ces s with stationary indep enden t incremen ts is the sum of n indep enden t v ariables X k ,n with a common distribution and hence Y ( λ ) has an infinitely div isible distribution . W e can summarise all ab ov e b y simply writing the c haracteristic function of Y ( λ ) for an y λ > 0 a s ϕ ( t, λ ) := I E  e itX ( λ )  = { ϕ ( t, 1) } λ . (2 . 5) W e no t e that w e hav e a do pted the notation commonly used in the early con tributions: the letter t denotes the F ourier parameter of the character- istic function whereas the contin uous parameter (essen tially the time) of a 5 sto c hastic pro cess has been denoted by the letter λ . Only later, when the theory of sto c hastic processes b ecame w ell dev elop ed, t he authors had de- noted the F ourier parameter b y a differen t letter like u or κ reserving, as natural, the letter t to the time en tering the sto c hastic pro cess es. The reader should b e aw are of the old notation in order to a v oid p ossible confusion. Let us close this section b y recalling (essen tially based on the b o ok by Luk acs) the main theorems concerning the structure of infinitely divisible distributions, that are relev an t to our historical surv ey . First de Finetti’s Theorem : A c haracteristic f unction is infinitely divisible iff it has the form ϕ ( t ) = lim m →∞ exp { p m [ ψ m ( t ) − 1] } , (2 . 6) where the p m are real p ositive n um b ers while ψ m ( t ) are c hara cteristic func- tions. Second de Finetti’s Theorem : The limit of a sequence of finite pro ducts of P oisson-t ype c haracteristic functions is infinitely divisible. The con v erse is also true. This means that the class of infinitely divisible laws coincide s with the class of distribution limits of finite conv olutio ns o f distributions of P oisson-t ype 4 . The K olmogoro v canonical r epresentation : The function ϕ ( t ) is the c haracteristic f unction of an infinitely divisible distribution with finite second momen t iff it can b e written in the form log ϕ ( t ) = iγ t + Z + ∞ −∞  e itu − 1 − itu  dK ( u ) u 2 , (2 . 7) where γ is a real constant, and K ( x ) is a non-decreasing and b ounded f unc- tion suc h that K ( −∞ ) = 0. The integrand is defined for u = 0 to be equal to − ( t 2 / 2). 4 Let us r ecall that for the characteristic function o f the Poisson distribution we have according to (2.4) ϕ ( t ) = exp  λ  e it − 1  , so that ϕ n ( t ) = exp  λ n  e it − 1   . The theorem can b e used to show that a given c haracter istic funct ion is infinitely divisible. F or an exa mple we r efer the reader to [80, p. 1 13]. 6 The L ´ evy canonica l represen tation : The function ϕ ( t ) is the c haracter- istic function o f an infinitely divisible distribution iff it can b e written in the form log ϕ ( t ) = iγ t − σ 2 2 t 2 + Z − 0 −∞  e itu − 1 − itu 1 + u 2  dM ( u ) + Z ∞ +0  e itu − 1 − itu 1 + u 2  dN ( u ) , (2 . 8) where γ is a real constan t, σ 2 is a real and non-negative constant, and the functions M ( u ), N ( u ) satisfy the follo wing conditions: (i) M ( u ) a nd N ( u ) are non-decreasing in ( −∞ , 0) and (0 , + ∞ ), resp ectiv ely . (ii) M ( −∞ ) = N (+ ∞ ) = 0 . (iii) The inte grals R 0 − ǫ u 2 dM ( u ) R + ǫ 0 u 2 dN ( u ) are finite fo r ev ery ǫ > 0. The L´ evy-Khin tc hine canonical represen tation : The function ϕ ( t ) is the c haracteristic function o f an infinitely divisible distribution iff it can b e written in the fo rm log ϕ ( t ) = iγ t + Z + ∞ −∞  e itu − 1 − itu 1 + u 2  1 + u 2 u 2 dG ( u ) , (2 . 9) where γ is a real constan t, and G ( u ) is a non- decreasing and b ounded function suc h that G ( −∞ ) = 0. The integrand is defined for u = 0 to b e equal to − ( t 2 / 2). W e p oint out that there is a tigh t connection b et w een the L ´ evy-Khintc hine canonical represen tation and the g eneral Cen tral Limit Theorem. F o r a clear description of a mo dern view on this connection w e refer, e.g. , to [49]. 3 The w ork of de Finetti and Kolmogor ov Bruno de Finetti is recognized to b e the most prominent scien tist of the Italian sc hoo l of Probability and Statistics, that started at the b eginning of the la st cen tury with Guido Casteln uo v o (1865-195 2 ) and F rancesco P ao lo Can telli (1875- 1966). His p ersonality and his in terest in probabilit y came out already with his attendance at the 1928 In ternational Congress of Math- ematicians 5 held in Bo lo gna (Italy) from 3 to 10 Septem b er 1928. The young 5 The Cha irman of the Cong ress was Salv ato r e Pincherle (1853- 1 936), Pr ofessor of Math- ematics at the Univ ersity o f Bo lo gna from 1880 up to 1928, the year of the Congr ess. He was the first Pres ident of the Unione Matematica Italiana (UMI) from 192 2 up to 1936, at his death. 7 de Finetti presen t ed a note on the role of the ch aracteristic function in ran- dom phenomena [33] 6 , that w as published only in 19 32 in the Pro ceedings of the Congress 7 . A t the Bologna Congress de Finetti ha d the o ccasion to meet L ´ evy and Khin tc hine (who w ere included in the F renc h and Russian delegations, respec- tiv ely) but w e are not informed abo ut their in teraction. W e note, how ev er, that Khin tc hine did not presen t any comm unication whereas L ´ evy presen ted a note outside the field o f probabilit y , precis ely o n fractional differen t ia tion [72]; furthermore Kolmogorov did not attend t he Congress. S urely L´ evy , Kolmogorov a nd Khintc hine held in high consideration the Italian sc ho ol of Probabilit y since in the thirties they submitted some relev an t pap ers to Ital- ian j ournals (written in Ita lian f or the Russians and in F renc h f o r L´ evy), see e.g. [59, 60, 61, 6 8 , 69, 73, 7 4]. Just after the Bologna Congress de F inetti started a researc h regard- ing functions with random incremen ts, see [26, 27, 28, 2 9, 31] based on the theory of infinitely divisible c hara cteristic functions, ev en if he did not use suc h term. His results can b e summarized in a n um b er of relev ant theorems (partly stated in the previous Section). As it w as a lready mentioned t hey are highly connected with the sto c hastic pro ces ses with statio nary indep en- den t incremen ts. In this resp ect w e refer the reader to the Section 2.2 of the excellen t pap er b y Cifarelli and Regazzini on de Finetti’s contributions in Probabilit y and Statistics [20]. The pa p ers by de Finetti, published in the p erio d 1929- 1931 (in Italian) in the Proceedings of the Ro y al Academ y of Lincei ( R endic onti del la R e al e A c c ademia Nazio nale dei Linc ei ) [26 , 27, 28, 29, 31], attracted the atten tion of Kolmogoro v who w as inte rested to solv e the so-called de Finetti’s prob- lem , that is the problem of finding the general form ula for the c haracteristic function of the infinitely divisible distributions. This pro blem was indeed attac k ed b y K olmogorov in 1 932 in tw o notes published in Italian in the 6 W e hav e to mention tha t this work by de Finetti was the first sig nificant contribution to the sub ject known now a s the theory of e x changeable sequences (of even ts). A more exhaustive acco unt app ear ed in 1931 in [3 0]. 7 The Pro ceedings were published by Zanichelli, Bologna, with all the details of the scientific a nd so cial progr ams, in 6 volumes, that app ear ed fro m 1929 to 1932. The pap ers, published in o ne of the following languages: Italian, F r ench, Ge r man and E nglish, were classified in 7 sessions according to their topic. The pa pe r s pres ent ed b y Cantelli [17], de Finetti, Romanovsky and Slutsky (Session IV, devoted to Actuaria l Sciences, Probability and Statistics) were included within the la st volume, published in 193 2. 8 same journal as de Finetti (Rendicon ti della Reale Accademia Nazionale dei Lincei), where he gav e an exhaustiv e answ er to de Finetti’s problem for the case of v aria bles with finite se c ond moment , see [68, 69]. These t w o notes are a v ailable in English in a unique pap er (No 13) in the Sele cte d Works of A.N. Kolmo gor ov with a commen t of V.M. Zo lo tarev, see [7 0]: the final result of Kolmogoro v is reported in Section 2 as Eq. (2.7), known as the Kolmogoro v canonical represen tation of the infinitely divisible ch arac- teristic functions. 4 The w ork of L ´ evy and Khin tc hine The general case of de Finetti’s problem, including also the case of infinite varianc e , was in v estigated in 1934-35 by L ´ evy [7 3, 74] who published t w o pap ers in F renc h in the Italian Journal: Annali della Reale Scuola Normale di Pisa. At that time L ´ evy w as quite intere sted in the so-called stable distri- butions tha t are kno wn to exhibit infinite v ariance, except for the particular case of the Gaussian. The approac h b y L ´ evy , w ell describ ed in his classical 1937 b o ok [75], is quite independent from that of Kolmogoro v, a s can b e understoo d from fo otnotes in his 1 934 pap er [73], t ha t w e r ep ort partly b elo w in original. F rom the fo ot note (1) w e learn that the results con t a ined in his pap er w ere presen t ed in three comm unications of the Academ y of Sciences (Comptes Rendus) of 26 F ebruary , 26 Marc h and 7 Ma y 1934. Then, in the fo otnote (6) , p. 339, the Author writes: [Ajout ´ e ´ a la c o rr e ction de s ´ epr euves] L e r´ esum´ e de ma note du 26 f´ ev ri e r, r´ edig´ e p ar M. Kolmo gor ov, a attir´ e mon attention sur deux Notes de M. B. de Finetti (see [26, 29]) et deux autr es de M. Kolmo gor ov lui-mˆ eme (see [68, 69]), publi´ ees dans les Atti A c c ademia Naz. Linc ei (VI ser). C es derni` er es notamment c ontiennent la solution du pr obl ` eme tr ait´ e d a ns l e p r ´ esent tr avail, dans le c as o ` u le pr o c essus est hom o g` ene et o` u la valeur pr ob able I E { x 2 } est finie. L e r ´ esultat fonda m ental du pr´ ese nt M ´ emoir e app ar ait donc c omme une extension d’un r´ esultat de M. Kolmo gor ov. This means that P . L ´ evy w as not a w are ab o ut the results on homogeneous pro cesses with indep enden t incremen ts obtained by B. de Finetti and b y A. N. Kolmogorov. The final result of L´ evy is rep o rted in Section 2 as Eq. (2 .8), kno wn as the L´ evy canonical representation of the infinitely divisible c haracteristic functions. 9 In a pap er of 193 7 Khintc hine [62] show ed that L´ evy’ s result can b e obtained also b y an extension of Kolmogorov ’s metho d: his final result, re- p orted in Section 2 as Eq. (2.9), is kno wn as the L´ evy-Khin tchine canon- ical represen tation of the infinitely divisible c haracteristic functions. The translation from the Russian of t his fundamental pap er can b e found in Ap- p endix. The theory of infinitely distributions w as then presen ted in German in the article [6 4 ] and in Russian in his 1938 b o ok on Limit Distributions for Sums of Indep endent R a ndom V ariab l e s [6 5 ]. Unfortunately , ma ny contributions b y Khin tc hine (b eing in Russian) re- mained almost unkno wn in the W est up t o the English translation of the treatise b y Gnedenk o and Kolmogorov [4 5] in 1954. The obituary of Khin tc hine [44], that G nedenk o (his former pupil) pre- sen ted at the 19 6 0 Berk eley Symp osium on Mathematical Statistics and Probabilit y , pro vides a g eneral des cription of the w orks of Khin tc hine along with a complete bibliography . F rom that w e learn that the 1938 b o ok by Khin tc hine was preceded b y a sp ecial course of lectures in Mosco w Unive rsit y that attra cted the in terest of A.A. Bobro v, D .A. Raik o v and B.V. Gnedenk o himself. Ac kno w ledgements The authors are gr a teful to R. Gorenflo and the anon ymous referees for useful commen ts. W e tha nk also O. Celebi for the help with the paper b y G.M. Ba wly published in T urk ey . App endix: Khin tc hine’s 1937 article A. Y a. Khin tc hine 8 : A new deriv atio n of a formula b y P . L ´ evy , Bul letin of the Mosc ow State University 1 (1937) 1 -5. A collection o f all the so-called infinitely divisible distributions w as disco v ered for the first time b y P . L ´ evy 9 . He has deriv ed a remark able f o rm ula for the logarithm of the c hara cteristic function of suc h a distribution. Because of 8 W e ha ve to remar k that the fo otnotes in this App endix ar e trans lation o f the or iginal ones by Khintc hine. 9 Ann. R. Scuola Norm. Pisa (Ser. II), 3 , pp. 33 7-366 (193 4). 10 the imp orta nce of this form ula I shall giv e here a new completely analytic and v ery simple pro of of it 10 . Let ϕ ( x ) be an infinitely divisible distribution and let ϕ ( t ) b e the cor- resp onding characteristic function. It is kno wn that for each h ≥ 0 the function ϕ ( t ) h is a characteris tic function as w ell. W e denote b y ϕ h ( x ) the corresp onding distribution. Thus log ϕ ( t ) = lim h → 0 ϕ ( t ) h − 1 h = lim h → 0 I h ( t ) , where I h ( t ) = 1 h + ∞ Z −∞  e itu − 1  dϕ h ( u ) . Put for eac h h > 0 G h ( u ) = u Z 0 v 2 v 2 + 1 dϕ h ( v ) h . ( A. 1) Clearly the function G h ( u ) is nondecreasing and b ounded. F urthermore, I h ( t ) = + ∞ Z −∞  e itu − 1  u 2 + 1 u 2 dG h ( u ) . T aking the real part of this for mula w e ha v e − Re I h ( t ) = + ∞ Z −∞ (1 − cos tu ) u 2 + 1 u 2 dG h ( u ) . ( A. 2) Let A h := Z | u |≤ 1 dG h ( u ) , B h := Z | u | > 1 dG h ( u ) , C h := A h + B h = + ∞ Z −∞ dG h ( u ) . Relation (A.2) gives us for t = 1 − Re I h (1) ≥ Z | u |≤ 1 (1 − cos tu ) u 2 + 1 u 2 dG h ( u ) ≥ cA h , ( A. 3) 10 The method of this pro of ca n be co nsidered as an extension of the idea by A. N. Ko l- mogorov. The la tter fo r med the base of the pro of of a n a na logous formula in the impo r tant case o f finite v ariance (see R endic onti dei Linc ei , 15 , pp. 805 -808 a nd 8 66-86 9 (1932 ). 11 where c is a strictly p ositiv e constant. In the same w a y for each t w e hav e − Re I h ( t ) ≥ Z | u |≥ 1 (1 − cos tu ) dG h ( u ) . Hence − 2 Z 0 Re I h ( t ) dt ≥ 2 B h − Z | u |≥ 1 sin 2 u u dG h ( u ) ≥ B h . ( A. 4) It follo ws from Eqs. (A.3)-(A.4 ) that C h = A h + B h = − Re I h (1) c − 2 Z 0 Re I h ( t ) dt . Since the function I h ( t ) uniformly con v erges on 0 ≤ t ≤ 2 as h → 0 to a finite limit, then C h is b ounded as h → 0 . Since G h (0) = 0, the functions G h ( u ) remain uniformly b ounded for h → 0. Therefore, there exists a seq uence of p ositive num b ers h n ( n = 1 , 2 , . . . ) suc h that h n → 0 as n → ∞ , and the seque nce of functions G h n ( u ) con v erges to a (b ounded nondecreasing) function G ( u ) as n → ∞ . With γ n = + ∞ Z −∞ dG h n ( u ) u (where the integral has a sense due to (A.1)), w e hav e log ϕ ( t ) = lim n →∞    itγ n + + ∞ Z −∞  e itu − 1 − itu 1 + u 2  u 2 + 1 u 2 dG h n ( u )    . Since the integrand of t he ab o v e integral is b ounded and con tin uous, this in tegral tends as n → ∞ to + ∞ Z −∞  e itu − 1 − itu 1 + u 2  u 2 + 1 u 2 dG ( u ) . Hence t he sequence γ n should conv erge to a certain p ositiv e constan t γ . Therefore, log ϕ ( t ) = itγ + + ∞ Z −∞  e itu − 1 − itu 1 + u 2  u 2 + 1 u 2 dG ( u ) . ( A. 5) 12 This is the P . L´ evy form ula up to certain unessen tial details concerning the w a y of its presen tation. T o prov e the uniqueness of the last represen tat io n it is easier to g et the in v ersion form ula. Let ∆( t ) = t +1 Z t − 1 log ϕ ( α ) dα − 2log [ ϕ ( t )] . Then Eq. (A.5) giv es immediately ∆( t ) = − 2 + ∞ Z −∞ e itu  1 − sin u u  dG (( u ) = + ∞ Z −∞ e itu dK ( u ) , where K ( u ) = − 2 u Z 0  1 − sin v v  dG (( v ) . Then, the we ll-kno wn P . L ´ evy in v ersion formula 11 yields K ( u ) = 1 2 π + ∞ Z −∞ 1 − e − itu it ∆( t ) dt . It follo ws t ha t K ( u ) (and hence G ( u )) is completely determined b y the func- tion ϕ ( t ). Then, w e can easily conclude that G h ( u ) → G ( u ) as h → 0 . If this w ere not true, then there it w o uld exist a sequen ce of f unctions G h ( u ) con v erging t o another function (differen t from G ( u )). This w ould giv e a n- other represen ta tion of the type (A.5 ) for the function log ϕ ( t ). Vice-v ersa, now we can sho w that if the loga r ithm of a function ϕ ( t ) is represen ted in the form (A.5) for a certain nondecreasing b ounded function G ( u ), then ϕ ( t ) is a characteristic function of an infinitely divisible distribu- tion. Let ε b e an a r bit r a ry p ositiv e num b er. Put ∆ ε = G ( ε ) − G ( − ε ) , G ε ( u ) =      G ( u ) , u ≤ − ε, G ( − ε ) , − ε ≤ u ≤ ε, G ( u ) − ∆ ε , u ≥ ε. 11 Calcul des pr ob abilit ´ es , Paris (19 25), p. 167 . In the general ca se the integral should be consider ed in the sense o f the Cauch y principa l v alue. 13 Since the function G ε ( u ) is b ounded and nondecreasing, w e can write G ε ( u ) = λ ε ϕ ε ( u ) , where λ ε is a p ositive num b er and the function ϕ ε ( u ) differs by an additiv e constan t from a certain distribution (this statemen t b ecomes trivial if the total v aria tion of G ε ( u ) is equal to zero for eac h ε > 0). Let further f ε ( t ) = Z | u | >ε  e itu − 1  dG ( u ) = + ∞ Z −∞  e itu − 1  dG ε ( u ) = = λ ε + ∞ Z −∞  e itu − 1  dϕ ε ( u ) = λ ε { ϕ ε ( t ) − 1 } , where ϕ ε ( t ) is the c haracteristic function of the distribution ϕ ε ( x ). Eviden tly , the expression λ ε n ϕ ε ( t ) + ( 1 − λ ε n ) is a c haracteristic function for each n ≥ λ ε . Hence the function n log ( λ ε n ϕ ε ( t ) + 1 − λ ε n !) = n log ( 1 + λ ε n [ ϕ ε ( t ) − 1] ) is the logarithm of a c haracteristic function. The same is true for its limit as n → ∞ whic h is equal to λ ε [ ϕ ε ( t ) − 1] = f ε ( t ) . Therefore, if G ( u ) is an arbitrary b ounded nondecreasin g function and ε is an arbitrary p ositiv e n um b er, then the in tegral Z | u | >ε  e itu − 1  dG ( u ) ( A. 6) is the logarithm of a certain c haracteristic function. The same is v alid also for the inte gral Z | u | >ε  e itu − 1 − itu 1 + u 2  dG ( u ) , 14 whic h differs from (A.6) only by a term itγ , where γ is a real constant. W e can also c hange in the la st in tegral dG ( u ) to (1 + u 2 ) /u 2 dG ( u ), since the function (1 + u 2 ) /u 2 is b o unded for | u | > ε . Finally w e can pass to the limit as ε → 0. Therefore the function lim ε → 0 Z | u | >ε  e itu − 1 − itu 1 + u 2  1 + u 2 u 2 dG ( u ) is the logarithm of a characteristic function. But the express ion (A.5) differs from this limit only by a term itγ and a term of the type − at 2 ( a ≥ 0) whic h is due to a p ossible discon tin uit y of G ( u ) at u = 0. Hence the function ϕ ( t ) is the pro duct of a c haracteristic function with an expression of the t yp e e itγ − at 2 , where γ is a real constan t and a ≥ 0 . The last expression is a c haracteristic function of a certain Gaussian La w. Hence the function ϕ ( t ) is a c haracteristic function as we ll. The corresp onding law is eviden tly infinitely divisible since λ log ϕ ( t ) is for eac h λ ≥ 0 the express ion of the same type as (A.5). Thus , by what is pro v ed ab o v e, λ log ϕ ( t ) is the lo garithm of a certain characteristic function. Supplemen t. B. V. Gnedenk o ha s p oin ted out that, to g et the statemen t for the expression preceding to (A.5), o ne needs to see that for α → ∞ the limit Z | u |≥ α dG h ( u ) → 0 is uniform with r esp e ct to h . 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