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Distributional equation is an important tool in the characterization theorybecause many characteristic properties of distributions can be transferred tosuch equations. Using a novel and natural approach, we retreat a remarkabledistributional equation whose corresponding functional equation in terms ofLaplace-Stieltjes transform is of the Poincare type. The necessary andsufficient conditions for the equation to have a unique distributional solutionwith finite variance are provided. This complements the previous results whichinvolve at most the mean of the distributional solution. Besides, more generaldistributional (or functional) equations are investigated as well.

Necessary and Sufficient Conditions for Unique Solution to Functional Equations of Poincare Type