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The suprema of infinitely divisible processes

Witold Bednorz, Rafał Martynek

Abstract

In this paper we complete the full characterization of the expected suprema of infinitely divisible processes. In particular, we remove the technical assumption called $H(C_{0},δ)$ condition and settle positively the conjecture posed by M. Talagrand.

The suprema of infinitely divisible processes

Abstract

In this paper we complete the full characterization of the expected suprema of infinitely divisible processes. In particular, we remove the technical assumption called condition and settle positively the conjecture posed by M. Talagrand.

Paper Structure

This paper contains 17 sections, 15 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Infinitely divisible processes
  3. Rosiński's representation and Poisson point process
  4. Brief history of the problem
  5. Overiew of the results
  6. Upper bound
  7. Strategy for the lower bound
  8. Proofs
  9. Proof of Inequality (1)
  10. Proof of Inequality (2)
  11. Proof of Inequality (3)
  12. Proof of Inequality (4)
  13. Proof of Inequality (5)
  14. Appendix
  15. Extension to the separable case
  16. ...and 2 more sections

Key Result

Theorem 2

Consider a countable set $T$ of measurable functions on $\Omega$ such that each $t\in T$ satisfies $\int_{\Omega}t^2\wedge 1 d\nu<\infty$ and assume that $0 \in T$. Let $X_t$ be as in (intr3). Then

Theorems & Definitions (17)

  • Definition 1
  • Example 1
  • Theorem 2
  • Lemma 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 2
  • Lemma 3
  • Theorem 6
  • ...and 7 more