### Abstract

Let X_{1}, ..., 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} X_{1} X_{2} ... X_{n} / sqrt(∑^{*} X_{i1}^{2} ... X_{i}n - _{1}^{2}) follows the same distribution as X_{1}, where ∑^{*} denotes the sum of over all permissible sequences of integers 1 ≤ i_{1} <i_{2} <⋯ <i_{n - 1} ≤ n.

Original language | English |
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Pages (from-to) | 1662-1665 |

Number of pages | 4 |

Journal | Statistics and Probability Letters |

Volume | 77 |

Issue number | 17 |

DOIs | |

Publication status | Published - Nov 2007 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Statistics and Probability

### Cite this

*Statistics and Probability Letters*,

*77*(17), 1662-1665. https://doi.org/10.1016/j.spl.2007.04.023

**A connection between self-normalized products and stable laws.** / Melnykov, Igor; Chen, John T.

Research output: Contribution to journal › Article

*Statistics and Probability Letters*, vol. 77, no. 17, pp. 1662-1665. https://doi.org/10.1016/j.spl.2007.04.023

}

TY - JOUR

T1 - A connection between self-normalized products and stable laws

AU - Melnykov, Igor

AU - Chen, John T.

PY - 2007/11

Y1 - 2007/11

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 2 <⋯ n - 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 2 <⋯ n - 1 ≤ n.

KW - Data transformation

KW - Random walk

KW - Rayleigh model

KW - Self-normalized product

KW - Stable law

KW - Symmetric distribution

UR - http://www.scopus.com/inward/record.url?scp=35348826100&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35348826100&partnerID=8YFLogxK

U2 - 10.1016/j.spl.2007.04.023

DO - 10.1016/j.spl.2007.04.023

M3 - Article

VL - 77

SP - 1662

EP - 1665

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 17

ER -