On the rate of convergence in limit theorems for random sums via Trotter-distance

Tran Loc Hung,Tran Thien Thanh

doi:10.1186/1029-242x-2013-404

Abstract

The main purpose of this paper is to establish some estimates for the rates of convergence in limit theorems for random sums of independent identically distributed random variables via Trotter-distance.MSC:60F05, 60G50, 41A25.

Highlights

Let {Xn, n ≥ } be a sequence of independent identically distributed random variables with mean E(Xn) = μ and D(Xn) = σ < +∞, n ≥
(For Nn = we set SNn = S = .) In Robbins [ ] gave sufficient conditions for the validity of the central limit theorem for normalized random sums of ( )
Since the appearance of the Robbin’s work, various limit theorems concerning the asymptotic behaviors for randomly indexed sums of independent random variables and rates of convergence, either in the central limit theorem for random sums or in the weak law of large numbers for random sums have been studied systematically

Summary

Introduction

It is worth pointing out that the mathematical tools have been used in the study of limit theorems for random sums review to date, including characteristic function, positive linear operators and probability metrics Results of this nature may be found in the works of Feller [ ], Renyi [ ], Butzer and Schulz [ ], Kirschfink [ ], Rychlick and Szynal [ ], Cioczek and Szynal [ ], Zolotarev [ ] and [ ], Hung [ ] and [ ]. It is to be noticed that the definition of the Trotter-distance in terms of ( ) is defined by Kirschfink in [ ] as follows dT (X, Y ; f ) := sup f (x + t) d(P – Q)(x) , t∈R R where P and Q are probability distributions of two random variables X and Y , respectively, and f ∈ CB(R) (see Kirschfink [ ] for more details). For every f ∈ CB(R), we have the following estimation: SNn ; X ; f Nn

For f

Proof We shall begin with showing that k