G. P. Chistyakov

Limit theorems in free probability theory. I

Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line ℝ+ and on the unit circle \( \mathbb{T} \) we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory.

Moderate deviations for student's statistic

For self-normalized sums, say Sn/Vn, under symmetry conditions we consider Linnik-type zones, where the ratio

On completeness of random exponentials in the Bargmann–Fock space

\mathbf{P}\{S_n/V_n\ge x\}/(1-\Phi(x))

ON BOUNDS FOR MODERATE DEVIATIONS FOR STUDENT'S STATISTIC ∗

converges to~1, and establish optimal bounds for remainder terms related to this convergence.

A New Asymptotic Expansion and Asymptotically Best Constants in Lyapunov's Theorem. III

We study the completeness/incompleteness properties of a system of exponentials EΛ={eπλz; λ∈Λ}, viewed as elements of the Bargmann–Fock space of entire functions. We assume that the index set Λ is a realization of a random point field in C (the support of a random measure). We prove that the properties are determined by the density of the field, i.e., by the mean number of the field points per unit area. We also discuss certain implications and motivations of our results, in particular, the jumps of the integrated density of states of the Landau Hamiltonian with the random potential, equal to the sum of point scatters.

Limit theorems in free probability theory II

Let

Arithmetic of probability laws

X_1,X_2,\dots

Limit distributions of Studentized means

be independent random variables with zero means and finite variances. In this paper we prove lower bounds for a Cramer-type large deviation theorem for self-normalized sums which imply that the bounds obtained by Jing, Shao, and Wang [{\em Ann. Probab.}, 31 (2003), pp. 2167--2215] are sharp.

Distribution of the shape of Markovian random words

A new asymptotic expansion is obtained in Lyapunovs central limit theorem for distribution functions of centered and normed sums of independent random variables which are not necessary identically distributed. It is applied to determine the asymptotically best constants in the Berry-Esseen inequality, thus solving problems about their optimal values raised by Kolmogorov and Zolotarev.

Independence of linear forms with random coefficients

AbstractBased on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line ℝ+ and on the unit circle