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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