On a certain class of infinitely divisible distributions

Fourier Analysis of Semistable Distributions

“…Details of all this may be put together from the monographs [7,16] and [17], for example; the terminology went through changes and consolidation since Lévy [11], ours agrees with that in [16]. The description in (1-4) is basically due to Lévy [11] and, without any terminology, was rediscovered in the generality above by Shimizu [18], Kruglov [10] and Mejzler [13]. For multivariate extensions, initiated by Lévy [11] himself, the reader is referred to Sato [17,Chap.…”

“…Thus the validity of formula (13) for (k, j + 1) will follow if we find a common integrable bound, again, at least for all h ∈ [−δ, δ], where δ > 0 is some, suitably chosen constant.…”

“…If, however, convergence in distribution of V n is permitted along all possible subsequences of N, then by a classical theorem of Khinchin every infinitely divisible distribution becomes a limit, so all infinitely divisible distributions have nonempty domains of partial attraction. As an interesting intermediate case, Kruglov [10] and Mejzler [13] proved that if convergence is restricted to subsequences {n k } of N that satisfy the "geometric" growth condition lim k→∞ n k+1 /n k = c for some c > 1, then the limiting distribution is either normal or semistable with some exponent α ∈ (0, 2), with that c in (2), and, conversely, the domain of such geometric partial attraction of any semistable law is nonempty. (Partial results were obtained somewhat earlier by Shimizu [18] and Pillai [15]; the four papers are apparently independent.)…”

Merging to semistable laws

“…It turned out that the following Char acterization Theorem is true: If (1.3) holds along a subsequence {k n } С N for which liminf^oo k n+ i/k n -с for some с G (l,oo), then the distribu tion G^^a-of V(ip u ф 2 , &) is in У* such that, in the case when the exponent of G^^o--Gr^^o is a < 2, the multiplicative period of the functions Mi and M 2 in (1.4) is the с from this growth condition on {k n }. Conversely, for every G^^o-£ У* there exists an F such that if Х ъ X 2 ,... are indepen dent random variables with the common distribution function F, then (1.3) holds along a subsequence {k n } С N for which An equivalent form of this theorem was proved by Kruglov [17] and Mej zler [21] in terms of the Levy type description of У*, while the present version was obtained by Megyesi [20] with an independent proof within the framework of the «probabilistic» or «quantile-transform» approach of Csorgo, Haeusler and Mason [7]- [8] and Csorgo [5] to domains of attraction and partial attraction.…”

“…The realization of an apparent significance of У* D У starts with a remark of Doeblin [13], without any elaboration or, for that matter, even a precise statement, to the effect that semistable laws arise in the limit in (1.3) if the normalizing constants A kn satisfy a geometric growth con dition. Thirty years later, Shimizu [23] and Pillai [22] came close while Kruglov [17] and Mejzler [21] fully achieved that realization, all four of them acting independently of one another. It turned out that the following Char acterization Theorem is true: If (1.3) holds along a subsequence {k n } С N for which liminf^oo k n+ i/k n -с for some с G (l,oo), then the distribu tion G^^a-of V(ip u ф 2 , &) is in У* such that, in the case when the exponent of G^^o--Gr^^o is a < 2, the multiplicative period of the functions Mi and M 2 in (1.4) is the с from this growth condition on {k n }.…”