Explicit fixed points of the smoothing transformation

2022/09/19 by Peyrière, Jacques
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2209.08872

Abstract

We deal with the equation Y \stackrel\rm d= (1)/(b) ∑_{1≤ j≤ N} W_jY_j, where the unknown is the distribution of Y, the variables in the right hand side are independent, the Y_j are equidistributed with Y, N is an integer valued random variable, and the W_j are equidistributed, nonnegative and of expectation~1. Usually a solution is obtained as the limit of a martingale. In some cases we give an explicit formula for the law of Y.

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