Abstract:
Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory.

Abstract:
Let $T_1,...,T_n$ denote free random variables. For two linear forms $L_1=\sum_{j=1}^n a_jT_j$ and $L_2=\sum_{j=1}^n b_jT_j$ with real coefficients $a_j$ and $b_j$ we shall describe all distributions of $T_1,...,T_n$ such that $L_1$ and $L_2$ are free. For identically distributed free random variables $T_1,...,T_n$ with distribution $\mu$ we establish necessary and sufficient conditions on the coefficients $a_j,b_j,\,j=1,...,n,$ such that the statements:\quad $(i)$ $\mu$ is a centered semicircular distribution; and $(ii)$ \, $L_1$ and $L_2$ are identically distributed ($L_1\stackrel{D}{=}L_2$); are equivalent. In the proof we give a complete characterization of all sequences of free cumulants of measures with compact support and with a finite number of non zero entries. The characterization is based on topological properties of regions defined by means of the Voiculescu transform $\phi$ of such sequences.

Abstract:
We prove an expansion for densities in the free CLT and apply this result to an expansion in the entropic free central limit theorem assuming a moment condition of order 8 for the free summands.

Abstract:
We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution. We apply these results to the expansion of densities to derive expansions for the free entropic distance of sums to the Wigner law.

Abstract:
Optimal stability estimates in the class of regularized distributions are derived for the characterization of normal laws in Cramer's theorem with respect to relative entropy and Fisher information distance.

Abstract:
Sharpened forms of the concentration of measure phenomenon for classes of functions on the sphere are developed in terms of Hessians of these functions.

Abstract:
Edgeworth-type expansions for convolutions of probability densities and powers of the characteristic functions with non-uniform error terms are established for i.i.d. random variables with finite (fractional) moments of order $s \geq 2$, where $s$ may be noninteger.

Abstract:
For regularized distributions we establish stability of the characterization of the normal law in Cramer's theorem with respect to the total variation norm and the entropic distance. As part of the argument, Sapogov-type theorems are refined for random variables with finite second moment.