Rates of convergence for tail series
type: Item
instance of: scholarly article
title: Rates of convergence for tail series
zbMATH Open document ID: 0815.60027
DOI: 10.1016/0378-3758(94)00009-K
published in: Journal of Statistical Planning and Inference
publication date: 1995-06-30T00:00:00Z
review text: Let \(X_ 1,X_ 2,\dots\) be independent random variables with partial sums \((S_ n)\). Assume that \(\sum^ \infty_{i = 1} X_ i\) is a.s. convergent, and define \(T_ n = \sum^ \infty_{i = n + 1} X_ i\) for \(n \geq 0\). Let \((a_ n)\) be a positive, nondecreasing, divergent real sequence. Define the numbers \[ \alpha = \inf\left\{r : \sum^ \infty_{i = 1} P(a_{n_ i} T_{n_ i} > r) < \infty\quad \text{for every admissible sequence } (n_ i)\right\}, \] \[ \beta = \inf\left\{r : \sum^ \infty_{i = 1} P(a_{n_ i}(S_{n_{i + 1}} - S_{n_ i}) > r) < \infty \quad \text{for every admissible sequence } (n_ i)\right\}. \] Suppose that \[ \liminf_{n \geq m \to \infty} P(a_ m(S_ n - S_ m) > -t)>0 \quad\text{for every } t>0. \] Then \[ \limsup_{n\to \infty} a_ n T_ n \leq \alpha \quad \text{a.s.},\tag{1} \] \[ \limsup_{n \to \infty} a_ n T_ n = \alpha = \beta \quad \text{a.s. if } \alpha < \infty.\tag{2.} \]
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 731933
zbMATH Keywords: patial sums
zbMATH Keywords: tail series
zbMATH Keywords: law of iterated logarithm
zbMATH Keywords: independent random variables
author: R. James Tomkins
MaRDI profile type: Publication
cites work: Q4765805
cites work: Strong Laws for Ruled Sums
cites work: On Necessary and Sufficient Conditions for the Law of the Iterated Logarithm
cites work: Q3790034
cites work: Some Iterated Logarithm Results Related to the Central Limit Theorem
cites work: Limit theorems without moment hypotheses for sums of independent random variables
full work available at URL: 0378-3758(94)00009-k
OpenAlex ID: W2090701262
description: scientific article; zbMATH DE number 731933