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Revisiting the identification problem of a function by the ratio of Laplace transforms of powers

Dongdong Hu, Linglong Yuan, Minzhi Zhao

Abstract

The ratio of Laplace transforms of powers of a function arises in the context of auction theory. The question whether a function is uniquely identified by this ratio has been answered affirmatively, if the function is non-negative, non-decreasing and right analytic.This paper extends the result to a larger class of functions without monotonicity.A conjecture in the literature says that all càdlàg functions can be identified by the ratio. We disprove this conjecture by providing simple functions that cannot be identified.

Revisiting the identification problem of a function by the ratio of Laplace transforms of powers

Abstract

The ratio of Laplace transforms of powers of a function arises in the context of auction theory. The question whether a function is uniquely identified by this ratio has been answered affirmatively, if the function is non-negative, non-decreasing and right analytic.This paper extends the result to a larger class of functions without monotonicity.A conjecture in the literature says that all càdlàg functions can be identified by the ratio. We disprove this conjecture by providing simple functions that cannot be identified.
Paper Structure (8 sections, 9 theorems, 57 equations, 1 figure)

This paper contains 8 sections, 9 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs of Theorem \ref{['theorem1.1']} and \ref{['theorem1.2']}
  4. Technical preparations
  5. Concatenated functions
  6. Proofs of Theorem \ref{['theorem1.1']} and \ref{['theorem1.2']}
  7. Acknowledgement
  8. Appendix: a special case

Key Result

Theorem 2.1

Suppose that $p$ and $q$ are two right analytic functions of exponential order defined on $[0,\infty)$, with $p(0)=1$, and one of the following conditions holds: Then if $H_{n,m}(p,\cdot)=H_{n,m}(q,\cdot)$, we have $p=q$.

Figures (1)

  • Figure 1: The shape of $f(x)$ for different choices of $n,m$.

Theorems & Definitions (20)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.1
  • ...and 10 more