Recursive integral equations for random weights averages: Exponential functions and Cauchy distribution

A.R. Soltani

Statistics & Probability Letters, 2022, vol. 190, issue C

Abstract: In this article, firstly, we prove that functions of the form ϕ(x)=ecxI(−∞,0)(x)+ebxI[0,+∞)(x), c,b constants, are the only solutions to the integral equation ϕ(x)=∫01ϕ(ux)ϕ((1−u)x)du. This indeed gives the result of Van Asshe (1987) who used the Schwartz distribution theory to prove that for i.i.d X and Y, UX+(1−U)Y=dX if and only if X has a Cauchy distribution. Secondly, by looking into certain recursive integral equations involving characteristic functions, we prove that if for an n≥2, the random weight mean U(1)X1+(U(2)−U(1))X2+⋯+(1−U(n−1))Xn has a Cauchy distribution, then X1 has a Cauchy distribution; random variables X1,…,Xn are i.i.d, the random weights are the cuts of (0,1) by a uniform sample. The multivariate analogue of this result is also provided.

Keywords: Exponential function; Recursive integral equations; Cauchy distribution; Random weight averages (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spl.2022.109606

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