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Teor. Veroyatnost. i Primenen., 1962, Volume 7, Issue 2, Pages 170–184 (Mi tvp4711)  

Some Theorems on Non-Latticed Random Walk

A. A. Borovkov

Novosibirsk

Abstract: Let $\xi _1,\xi_2,…$ be identically distributed independent non-latticed random variables with a finite mean and a finite variance if ${\mathbf M}\xi_k=0$. Formulas are derived for the distribution of the first jump over the level $x,0\leq x\leq\infty$. In particular the following is proved: if $\chi _+(\chi_-)$ denotes the first positive (negative) sum, $\zeta=\inf(0,\xi _1+\xi _2+\cdots+\xi _n)$ and $p=P(\zeta=0)$, then
$$\frac{1-\mathbf{M}e^{i\lambda\xi _1}}{-2^{-1}\mathbf{D}\xi_1}=\frac{1-\mathbf{M}e^{i\lambda\chi_+}}{\mathbf{M}_{\chi_+}}\cdot\frac{1-\mathbf{M}e^{i\lambda\chi_-}}{\mathbf{M}\chi_-},\qquad{when}\qquad\mathbf{M}\xi _1=0,$$
$$\frac{1-\mathbf{M}e^{i\lambda\xi _1}}{\mathbf{M}\xi_1}=\frac{1-\mathbf{M}e^{i\lambda\chi_+}}{\mathbf{M}_{\chi_+}}\cdot\frac{1+p-\mathbf{M}e^{i\lambda\chi_-}}{p},\qquad{when}\qquad\mathbf{M}\xi _1>0.$$

Full text: PDF file (1388 kB)

English version:
Theory of Probability and its Applications, 1962, 7:2, 164–179

Received: 07.04.1960

Citation: A. A. Borovkov, “Some Theorems on Non-Latticed Random Walk”, Teor. Veroyatnost. i Primenen., 7:2 (1962), 170–184; Theory Probab. Appl., 7:2 (1962), 164–179

Citation in format AMSBIB
\Bibitem{Bor62}
\by A.~A.~Borovkov
\paper Some Theorems on Non-Latticed Random Walk
\jour Teor. Veroyatnost. i Primenen.
\yr 1962
\vol 7
\issue 2
\pages 170--184
\mathnet{http://mi.mathnet.ru/tvp4711}
\transl
\jour Theory Probab. Appl.
\yr 1962
\vol 7
\issue 2
\pages 164--179
\crossref{https://doi.org/10.1137/1107015}


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