Convergence in Mallows distance is of particular interest when heavy-tailed distributions are considered. For 1≦α<2, it constitutes an alternative technique to derive central limit type theorems for non-Gaussian α-stable laws. In this note, we further explore the connection between Mallows distance and convergence in distribution. Conditions for their equivalence are presented.
Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations
of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any
one of n cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms
and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic
distributions for the total winnings of the n players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For
all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that
for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic
expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be
numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations
even for very small n.
For the derivativesp(k)(x; α, γ) of the stable density of index α asymptotic formulae (of Plancherel Rotach type) are computed ask→∞ thereby exhibiting the detailed analytic structure for large orders of derivatives. Generalizing known results for the
special case of the one-sided stable laws (O<α<1, γ=-α) the whole range for the index of stability and the asymmetry parameter γ is covered.
Authors:J. Domínguez-Molina and Alfonso Rocha-Arteaga
The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible
distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under
this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models
considered in Benaych-Georges  who introduced the models through their laws. Second, random matrix models for free infinitely
divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding
classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations
of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix
models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free
selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ∫0∞e−1dΨtd, d ≥ 1, where Ψtd is a d × d matrix-valued Lévy process satisfying an Ilog condition.