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A local limit theorem for nonlattice multidimensional random walks in cones

Thi da Cam Pham, Marc Peigné, Doan Thai Son

Abstract

We study the asymptotic behavior of a nonlattice random walk in a general cone of $R^d$ . Following the approach initiated by D. Denisov and V. Wachtel in [8], we use a strong approximation of random walks by the Brownian motion and prove local limit theorems, combining integral theorems for random walks in cones with classical theorems for unrestricted random walks.

A local limit theorem for nonlattice multidimensional random walks in cones

Abstract

We study the asymptotic behavior of a nonlattice random walk in a general cone of . Following the approach initiated by D. Denisov and V. Wachtel in [8], we use a strong approximation of random walks by the Brownian motion and prove local limit theorems, combining integral theorems for random walks in cones with classical theorems for unrestricted random walks.

Paper Structure

This paper contains 10 sections, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Random walks in cones
  4. Main results
  5. Preparatory results
  6. Local Riemann integrability of the harmonic function $V$
  7. Reversing time and key approximation lemma
  8. Some useful estimates
  9. Proof of Theorem \ref{['theostonecone']}
  10. Proof of Theorem \ref{['theolocal']}

Figures (1)

  • Figure 1: Cones ${\color{blue} \mathcal{C}_{\delta}}\subset \mathcal{C}\subset {\color{red} \mathcal{C}_{-\delta}}$