Does the ratio of Laplace transforms of powers of a function identify the function?

Konstantopoulos, Takis and Yuan, Linglong ORCID: 0000-0002-7851-1631 Does the ratio of Laplace transforms of powers of a function identify the function?

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Abstract

We study the following question: if $f$ is a nonzero measurable function on $[0,\infty)$ and $m$ and $n$ distinct nonnegative integers, does the ratio $\widehat{f^n}/\widehat{f^m}$ of the Laplace transforms of the powers $f^n$ and $f^m$ of $f$ uniquely determine $f$? The answer is yes if one of $m, n$ is zero, by the inverse Laplace transform. Under some assumptions on the smoothness of $f$ we show that the answer in the general case is also affirmative. The question arose from a problem in economics, specifically in auction theory where $f$ is the cumulative distribution function of a certain random variable. This is also discussed in the paper.

Item Type:	Article
Uncontrolled Keywords:	math.PR, math.PR, Primary: 44A10, 26Axx, Secondary: 91B70, 91B26
Depositing User:	Symplectic Admin
Date Deposited:	12 Sep 2019 07:53
Last Modified:	19 Jan 2023 00:26
Related URLs:	Author
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3054278

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• Does the ratio of Laplace transforms of powers of a function identify the function? (deposited 12 Sep 2019 07:53) [Currently Displayed]
  • Does the ratio of Laplace transforms of powers of a function identify the function? (deposited 21 Sep 2020 10:02)