Some Theorems on Non-Latticed Random Walk

A. A. Borovkov

Teoriâ veroâtnostej i ee primeneniâ, Tome 7 (1962) no. 2, pp. 170-184

Résumé

Let $\xi _1,\xi_2,\dots$ 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 $$\frac{1-\mathbf{M}e^{i\lambda\xi _1}}{-2^{-1}\mathbf{D}\xi_1}=\frac{1-\mathbf{M}e^{i\lambda\chi_+}}{\mathbf{M}_{\chi_+}}\cdot\frac{1-\mathbf{M}e^{i\lambda\chi_-}}{\mathbf{M}\chi_-},\qquad{\text{when}}\qquad\mathbf{M}\xi _1=0,$$$$\frac{1-\mathbf{M}e^{i\lambda\xi _1}}{\mathbf{M}\xi_1}=\frac{1-\mathbf{M}e^{i\lambda\chi_+}}{\mathbf{M}_{\chi_+}}\cdot\frac{1+p-\mathbf{M}e^{i\lambda\chi_-}}{p},\qquad{\text{when}}\qquad\mathbf{M}\xi _1>0.$$