Convergence of some functions of the mean value of independent random variables. (Russian) Zbl 0695.60028

Grigor’yan, A. G.

Teor. Veroyatn. Mat. Stat., Kiev 41, 33-36 (1989).

Let \(\{X_ k\), \(k\geq 1\}\) be a sequence of i.i.d. random variables with finite expectation \(\mu\) and variance \(\sigma^ 2\). Assume that a real function g is m times differentiable with \(g^{(m)}(\mu)\neq 0\), \(g^{(k)}(\mu)=0\), \(1\leq k\leq m-1\). The limit of \[ P\{m!(n)^ m[g^{(m)}(\mu)\sigma^ m]^{-1}[g(S_ n/n)-g(\mu)]<x\} \] and the rate of convergence to the limit is investigated when \(n\to \infty\).

Reviewer: D.Szynal

MSC:

60F05

Central limit and other weak theorems

Keywords:

rate of convergence