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On the Wiener-Hopf factorization for one class of random walk. (English. Russian original) Zbl 0910.60059
Dokl. Math. 53, No. 3, 355-357 (1996); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 348, No. 2, 162-164 (1996).
The sequence $$(X_n)^\infty_1$$ of i.i.d. random variables is considered. The ladder variable $$\chi_+ =S_{\eta_+}$$ is investigated, where $$\eta_+ =\inf \{n\geq 1; S_n>0\}$$, and $$S_n= X_1+ \cdots+X_n$$. Let $$r_{z+} (\lambda)= 1-E(z^{\eta_+} e^{\lambda\chi_+}; \eta_+< \infty)$$, where $$z,\lambda$$ are complex values. Assume $$1-z Ee^{\lambda X_1}= g_z (\lambda)/h_z (\lambda)$$, where $$g_z$$, $$h_z$$ are entire functions of finite order having no common zeros. A representation of $$r_{z_+}$$ in terms of these zeros is proved as a corollary of some known theorem on entire functions. This representation has implicit view when the distribution of $$X_1$$ is Gaussian with the mean value $$a$$ and the variance $$\sigma^2$$. In particular $r_{1_+} \left( {\lambda \over\sigma} \right)= -{\lambda \over \sqrt 2}\exp \left({K \over\sqrt {2\pi}} \lambda+ {\lambda^2 \over 8} \right) \bigcap^\infty_{n=1} {1\over 2} \left(\left(1-{\lambda \over \sqrt {2\pi n}} \right)^2+1 \right) \exp {\lambda \over \sqrt {2\pi n}},$ where $$K= 1.460\dots$$.
##### MSC:
 60G50 Sums of independent random variables; random walks 60J35 Transition functions, generators and resolvents