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Short Communications

On characteristic functions of probability distributions of sums with random permutations of signs

A. A. Ryabinin

N. I. Lobachevski State University of Nizhni Novgorod, Faculty of Mechanics and Mathematics

Abstract: In the paper, the random series
$$ S=\sum_{k=1}^\infty \pm a_k ,\qquad a_k > 0,\qquad \sum_{k=1}^\infty a_k < \infty $$
$S=\sum_{k=1}^\infty \pm a_k$, $a_k > 0$, $\sum_{k=1}^\infty a_k < \infty$ is considered, in which the permutation of signs is subject to the Markov dependence with the matrix of transition probabilities
$$ \begin{pmatrix} p(+1,+1)&p(-1,+1) p(+1,-1)&p(-1,-1) \end{pmatrix}= \begin{pmatrix} 1-\alpha&\alpha \alpha&1-\alpha \end{pmatrix}, \qquad 1<\alpha<1. $$
For the characteristic function $f(z)$ of the sum $S$, the formula
$$ f(z)=\prod^{\infty}_{k=0}\cos(a_kz)+i(1-2\alpha)\sum_{j=0}^{\infty}\psi_j(z)\prod^{\infty}_{k=j+2}\cos(a_kz)\sin(a_{j+1}z), $$
is obtained, where $\psi_j(z)=\mathsf{E}(t_je^{izS_j})$ и $S_j=\sum^j_{k=1}\pm a_k$, $z \in {\mathbf C}^1$.

Keywords: random series, Markov dependence, characteristic function.

Received: 12.04.1999

DOI: 10.4213/tvp508

 English version: Theory of Probability and its Applications, 2001, 45:4, 687–690

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