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Recursive integral equations for random weights averages: Exponential functions and Cauchy distribution by A.R. Soltani is a Mathematics article available to read on EtoBox.
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In this article, firstly, we prove that functions of the form φ(x) = e cx I (-∞,0) (x) + e bx I [0,+∞) (x), c, b constants, are the only solutions to the integral equation φ(x) = ∫ 1 0 φ(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 , UXonly 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 meanX n has a Cauchy distribution, then X 1 has a Cauchy distribution; random variables X 1 , . . . , X n 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.
Who reads Recursive integral equations for random weights averages: Exponential functions and Cauchy distribution?
It is typically read by researchers, students, and practitioners in Mathematics.
- Author
- A.R. Soltani
- Publisher
- Elsevier BV
- Published
- 2022
- Language
- EN
- Field
- Mathematics (Physical Sciences)