Characterization of discrete laws via mixed sums and Markov branching processes (Q1346160)

Equations of the form \[ X{\buildrel{\text{d}}\over =} U(X_ 1 + X_ 2), \tag{1} \] with \(X_ j {\buildrel{\text{d}} \over =} X\) and \(U\) independent of \((X_ 1, X_ 2)\), and uniformly distributed on (0,1), were first considered by Shanbag [see \textit{M. H. Alamatsaz}, Acta Math. Hung. 45, 159-162 (1985; Zbl 0601.60011)]. There have been many generalizations and extensions of (1), especially to integer-valued random variables. The present paper gives the most general and most detailed results. The author considers the equation \[ C_ u {\buildrel{\text{d}}\over =} Q \otimes C_{u+v}, \tag{2} \] where \(C\) is a \(\mathbb{Z}_ +\)-valued process with stationary and independent increments, either in discrete or in continuous time; \(Q\) is a r.v. with values in \([0,1]\), independent of \(C\). The operator \(\otimes\) was introduced by \textit{K. van Harn}, the reviewer and \textit{W. Vervaat} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 61, 97-118 (1982; Zbl 0476.60016)], and is defined in terms of branching processes. Equation (2) translates into a nonlinear integral equation for the p.g.f. \(g(s)\), of \(C_ 1\), which is studied in terms of regular varying function of \((1 - s)^{-1}\). Conditions for existence and uniqueness of solutions are obtained, and relations between solutions of the discrete-time and continuous-time problems are exhibited. Some explicit solutions are given. Partial overlap is indicated with results by \textit{K. van Harn} and the reviewer [Stochastic Processes Appl. 45, No. 2, 209-230 (1993; Zbl 0773.60081)]. An attempt is made to obtain two-sided solutions of (2). This is only possible in special cases, and the problem seems to be less natural; even in the single case (1), solutions are not unique.