### Book chapter

## Edgeworth Corrections in Randomized Central Limit Theorems

We consider rates of approximation of distributions of weighted sums of independent, identically distributed random variables by the Edgeworth correction of the 4-th order.

### In book

Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of modulus 1. The functions γ we consider are the corresponding eigenfunctions. In Theorem 1.1, we prove that the limit inferior of the ergodic sums (n,γ(x_0)+⋯+γ(x_{n−1})) n∈N is bounded for every point x in the phase space. In Theorem 1.2, we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length. Under additional assumptions, we prove that ergodic integrals satisfy the Central Limit Theorem (Theorems 1.3 and 1.10).

Digital technologies provide new possibilities for studying cultural heritage. Thus, literature research involving large text corpora allows to set and solve theoretical problems which previously had no prospects for their decision. For example, it has become possible to model the literary system for some defi-nite literary period (i.e., for the Silver Age of Russian literature) and to classify prose writers according to their stylistic features. And more than that, it allows to solve more general theoretical problems. The given research was conducted on Russian literary texts of the early 20th century. The sample included 100 short stories by 100 different writers. The measurements were carried out for 5 syntactic variables. For each of these distributions, the most popular statistics were calculated. Basing on these data, we consider empirical verification of Lyapunov's central limit theorem (CLT). The article validates the effectiveness of CLT theorem and the conditions for its implementation. Besides the normal (Gaussian) function we used another analytical model — the Hausstein func-tion. It turned out that both theoretical distributions for each of five variables do not contradict the experimental data. However, the alternative analytical model (Hausstein function) has shown even better agreement with the experimental data. The obtained results may be used in computational linguistic studies and for research of Russian literary heritage.

We consider the typical behavior of the weighted sums of independent identically distributed random vectors in k-dimensional space. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order O(1/n) up to logarithmic factor. This extends the one-dimensional Klartag and Sodin result.

In this paper we consider the product of two independent random matrices X^(1) and X^(2). Assume that X_{jk}^{(q)},1\le j,k \le n,q=1,2,, are i.i.d. random variables with \EX_{jk}^{q}=0, VarX_{jk}^{(q)}=1/ Denote by s_1(W),…,s_n(W) the singular values of W:=n^{-1}X^(1)X^(2). We prove the central limit theorem for linear statistics of the squared singular values s_1^2(W),…,s_n^2(W) showing that the limiting variance depends on \kappa_4:=\E(X_{11}^{(1)})^4−3.