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

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

1 Scopus citations

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 (US)
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
State	Published - Nov 2007
Externally published	Yes

Keywords

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

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10.1016/j.spl.2007.04.023

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title = "A connection between self-normalized products and stable laws",

abstract = "Let X1, ..., Xn 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 n1 / 2 α X1 X2 ... Xn / sqrt(∑* Xi12 ... Xin - 12) follows the same distribution as X1, where ∑* denotes the sum of over all permissible sequences of integers 1 ≤ i1 < i2 < ⋯ < in - 1 ≤ n.",

keywords = "Data transformation, Random walk, Rayleigh model, Self-normalized product, Stable law, Symmetric distribution",

author = "Igor Melnykov and Chen, {John T.}",

year = "2007",

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AU - Chen, John T.

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N2 - Let X1, ..., Xn 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 n1 / 2 α X1 X2 ... Xn / sqrt(∑* Xi12 ... Xin - 12) follows the same distribution as X1, where ∑* denotes the sum of over all permissible sequences of integers 1 ≤ i1 < i2 < ⋯ < in - 1 ≤ n.

AB - Let X1, ..., Xn 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 n1 / 2 α X1 X2 ... Xn / sqrt(∑* Xi12 ... Xin - 12) follows the same distribution as X1, where ∑* denotes the sum of over all permissible sequences of integers 1 ≤ i1 < i2 < ⋯ < in - 1 ≤ n.

KW - Data transformation

KW - Random walk

KW - Rayleigh model

KW - Self-normalized product

KW - Stable law

KW - Symmetric distribution

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ER -

A connection between self-normalized products and stable laws

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Keywords

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