\(p\)-adic behaviour of Bernoulli probabilities
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.
Bernoulli scheme for independent random variables
scientific article; zbMATH DE number 1160169