### On the rates of convergence to symmetric stable laws for distributions of normalized geometric random sums

#### Abstract

Let $X_{1}, X_{2}, \cdots$ be a sequence of independent and

identically distributed random variables.

Let $\nu_{p}$ be a geometric random variable with

mean $1/p, p\in (0, 1), $ independent of all $X_{j}, j\geq 1.$

Assume that $\varphi:\mathbb{N}\to\mathbb{R}^{+}$ is a

positive normalized function such that $\varphi (n)\downarrow 0$

as $n\to+\infty.$ The main purpose of this paper is to

establish the rates of convergence for distributions of normalized

geometric random sums

$\varphi (\nu_{p})\sum\limits_{j=1}^{\nu_{p}}X_{j}$

to symmetric stable laws via Zolotarev's probability metric.

identically distributed random variables.

Let $\nu_{p}$ be a geometric random variable with

mean $1/p, p\in (0, 1), $ independent of all $X_{j}, j\geq 1.$

Assume that $\varphi:\mathbb{N}\to\mathbb{R}^{+}$ is a

positive normalized function such that $\varphi (n)\downarrow 0$

as $n\to+\infty.$ The main purpose of this paper is to

establish the rates of convergence for distributions of normalized

geometric random sums

$\varphi (\nu_{p})\sum\limits_{j=1}^{\nu_{p}}X_{j}$

to symmetric stable laws via Zolotarev's probability metric.

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