Natural estimates of the accuracy of approximation of the distributions of random sums by location mixtures of stable laws (Q1781710)

The authors construct several estimates of the rate of convergence in the following theorem and special case of it: Let \(S_{N_n}:=\sum_{i=1}^{N_n}X_i\) and \(Z_n:=(S_{N_n}-an)/b(n)n^{1/\alpha},\) where \(N_n\) are integer-valued, nonnegative random variables independent of i.i.d. r.v. \(X_n, \quad n=1,2,3,\dots,\) \(b(n)\) is a slowly varying function and \(\alpha \in (1,2].\) Let also \[ N_n\to^{\text P}_{n\to+\infty}+\infty\text{ and }\left.\left(\sum_{j=1}^nX_j-na\right)\right/b(n)n^{1/\alpha}\to_{n\to+\infty}^{\text{weakly}}Y_{\alpha}, \] where \(Y_{\alpha}\) has distribution function \(G_{\alpha}(x), a\in R.\) Then \(Z_n\to^{\text{weakly}}_{n\to+\infty}Z\) iff there exists a random variable \(V\) such that \(Z=^{d}Y_{\alpha}+V,\) the random variables on the latter right-hand side are independent and \(a(N_n-n)/b(n)n^{1/\alpha}\to^{\text{weakly}}_{n\to+\infty} V.\) Some alternative approximations of one-dimensional distributions of random sums are considered. The rate of convergence is considered in terms of probability metrics that metrize weak convergence.