A connection between self-normalized products and stable laws

Igor Melnykov; John T. Chen

doi:10.1016/j.spl.2007.04.023

A connection between self-normalized products and stable laws

Igor Melnykov, John T. Chen

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Let X₁, ..., X_n constitute a random sample from a population with underpinning cumulative distribution function F (x). For any value 0 < α < 1, we prove that under a condition of stable laws, the self-normalized product n^{1 / 2 α} X₁ X₂ ... X_n / sqrt(∑^* X_i1² ... X_in - ₁²) follows the same distribution as X₁, where ∑^* denotes the sum of over all permissible sequences of integers 1 ≤ i₁ < i₂ < ⋯ < i_{n - 1} ≤ n.

Original language	English
Pages (from-to)	1662-1665
Number of pages	4
Journal	Statistics and Probability Letters
Volume	77
Issue number	17
DOIs	https://doi.org/10.1016/j.spl.2007.04.023
Publication status	Published - Nov 1 2007

Keywords

Data transformation
Random walk
Rayleigh model
Self-normalized product
Stable law
Symmetric distribution

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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KW - Symmetric distribution

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