Random walks and mixtures of gamma distributions (Q2882295)

The purpose of this paper is to expand the area where the Wiener-Hopf integral equation can effectively be solved; it considers the case when the kernel \(K\) is represented as a finite two-sided mixture of gamma distributions of the form NEWLINE\[NEWLINE K^{\pm}(x)=\sum_{j=1}^J\exp(-s_j^{\pm}x)P^{\pm}_{N,j}(x), NEWLINE\]NEWLINE where \(s_j^{\pm}\) and \(P_{N,j}^{\pm}\) are polynomials of order less than or equal to \(N\) with nonnegative coefficients NEWLINE\[NEWLINE P_{N,j}^{\pm}(x)=\sum_{m=0}^Na_{m,j}^{\pm}\frac{(s_j^{\pm})^{m+1}}{m!}x^m,\;a_{m,j}^{\pm}\geq 0,\;m=0,1,\ldots ,N;\;j=1,\ldots ,J. NEWLINE\]NEWLINE The authors describe in detail the case of a symmetric density with one exponential \(K(x)=\exp(-s|x|)P_N(x)\), and where \(P_N\) is a polynomial with nonnegative coefficients: NEWLINE\[NEWLINE P_N(x)=\sum_{m=0}^Na_m\frac{s^{m+1}}{m!}x^m,\;a_N>0,\;a_m\geq 0,\;m=0,1,\ldots ,N-1;\;s>0. NEWLINE\]NEWLINE The method of nonlinear factorization equations is applied, which has a direct connection with the known factorization identity in the theory of stochastic processes.