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Asymptotics of solutions to the linear search problem

Robin A. Heinonen

Abstract

The exact leading asymptotics of solutions to the symmetric linear search problem are obtained for any positive probability density on the real line with a monotonic, sufficiently regular tail. A similar result holds for densities on a compact interval.

Asymptotics of solutions to the linear search problem

Abstract

The exact leading asymptotics of solutions to the symmetric linear search problem are obtained for any positive probability density on the real line with a monotonic, sufficiently regular tail. A similar result holds for densities on a compact interval.
Paper Structure (14 sections, 13 theorems, 112 equations)

This paper contains 14 sections, 13 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem statement and known results
  4. The hazard function
  5. Regular variation and de Haan's class $\Gamma$
  6. Results
  7. Extension to compact intervals
  8. Examples
  9. Proof of Theorem \ref{['thm:increments']}
  10. Proof of Theorem \ref{['thm:asymptotics']}
  11. Proof of Proposition \ref{['prop:pareto']}
  12. Proof of Theorem \ref{['thm:bounded']}
  13. Proof of Proposition \ref{['prop:bounded_powerlaw']}
  14. Discussion

Key Result

Proposition 1

Let $\{x_k\}$ be a sequence of turning points parametrizing a minimizing search strategy for Problem prob:lsp. Then the following hold:

Theorems & Definitions (38)

  • Proposition 1
  • Definition 1
  • Example 1: Pareto
  • Example 2: Stretched/compressed exponential
  • Example 3: Lognormal
  • Example 4: Gumbel
  • Lemma 1
  • proof
  • Definition 2: Slow variation
  • Definition 3: Regular variation
  • ...and 28 more