\(p\)-adic behaviour of Bernoulli probabilities
type: Item
instance of: scholarly article
title: \(p\)-adic behaviour of Bernoulli probabilities
zbMATH Open document ID: 0908.60017
DOI: 10.1016/S0167-7152(97)00140-5
published in: Statistics \& Probability Letters
publication date: 1999-03-09T00:00:00Z
review text: Consider the standard Bernoulli scheme for independent random variables \(\xi_j (\omega) =0\) or 1 with probabilities 1/2 for \(j=1,2, \dots\). Let \(S_n (\omega): =\sum^n_{k=1} \xi_k (\omega)\) and \[ A(p,n): =\bigl\{\omega \in\{0,1\}^n: p\text{ divides } S_n (\omega) \bigr\}, \] where \(p\) is a prime number. Then \(P(A(p,n)) =\# A(p,n)/2^n\). Denote by \(\overline A(p,n)\) the complement of \(A(p,n)\). The author proves that \(A(p, n)\) and \(\overline A(p,n)\) are asymptotically symmetric: \(P(A_n (p,n)) \to 1/2\) and \(P (\overline A_n (p,n)) \to 1/2\) in the \(p\)-adic metric as \(n\) approaches 1 in the same metric.
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 1160169
zbMATH Keywords: Bernoulli scheme for independent random variables
zbMATH Keywords: asymptotically symmetric
author: Andrei Khrennikov
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
cites work: Q4003475
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cites work: \(p\)-adic probability interpretation of Bell's inequality
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description: scientific article; zbMATH DE number 1160169