On characteristic functions of probability distributions of sums with random permutations of signs
Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 4, pp. 773-776
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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.
@article{TVP_2000_45_4_a12,
author = {A. A. Ryabinin},
title = {On characteristic functions of probability distributions of sums with random permutations of signs},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {773--776},
year = {2000},
volume = {45},
number = {4},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2000_45_4_a12/}
}
TY - JOUR AU - A. A. Ryabinin TI - On characteristic functions of probability distributions of sums with random permutations of signs JO - Teoriâ veroâtnostej i ee primeneniâ PY - 2000 SP - 773 EP - 776 VL - 45 IS - 4 UR - http://geodesic.mathdoc.fr/item/TVP_2000_45_4_a12/ LA - ru ID - TVP_2000_45_4_a12 ER -
A. A. Ryabinin. On characteristic functions of probability distributions of sums with random permutations of signs. Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 4, pp. 773-776. http://geodesic.mathdoc.fr/item/TVP_2000_45_4_a12/