Random difference equations with logarithmic distribution and the triggered shot noise (Q1866182)

The limit in law of the sequence of random variables \(U_n\), \(n \geq 1\), such that \(U_0=0\), \(U_n=X_n (Y_n+U_{n-1})\), \(n\geq 1\), where \(X_i\) are i.i.d. random variables with density \(-\ln x\) on \([0,1]\) or \((-1/2)\ln x\) on \([-1,1]\), is studied. The authors consider the following five cases of distribution: deterministic, binary, Laplace, normal and Cauchy. The presented method may also be used for estimation of the suitable stationary densities.