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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 \(K= 1.460\dots\).
60G50 Sums of independent random variables; random walks
60J35 Transition functions, generators and resolvents