On integral transforms of distribution functions (Q1822295)

The authors are interested in the study of equations of the form \(f^ 1_{X\circ Y}(s)=f^ 2_ X*f^ 3_ Y.\) Here X and Y are random variables, one dimensional, on a subset of the reals. The functions \(f^ k\) are integral transforms, with unknown kernels \(G^ k\), of the functions \(F^ k\). That is, \(f^ k_ X(s)=\int_{A}G^ k(x,s)dF^ k(x).\) The operations \(\circ\) and * are also unknown. First, under suitable hypotheses, the system of solutions is obtained for the case of the unit step functions \(F^ k\). Then the general case is studied. A number of very nice illustrative examples are included which help to clarify the results. These examples include a set of cases in which the two operations are specified and then the kernels \(G^ k\) are obtained from the general results. Cases in which \(G^ 1=G^ 2=G^ 3\) are included.