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Renewal theorems in a periodic environment

Quentin Cormier

Abstract

We study a renewal problem within a periodic environment, departing from the classical renewal theory by relaxing the assumption of independent and identically distributed inter-arrival times. Instead, the conditional distribution of the next arrival time, given the current one, is governed by a periodic kernel, denoted as $H$. The periodicity property of $H$ is expressed as $\mathbb{P}(T_{k+1} > t ~ |~ T_k) = H(t, T_k)$, where $H(t+T,s+T) = H(t, s)$. For a fixed time $t$, we define $N_t$ as the count of events occurring up to time $t$. The focus is on two temporal aspects: $Y_t$, the time elapsed since the last event, and $X_t$, the time until the next event occurs, given by $Y_t = t - T_{N_t}$ and $X_t = T_{N_{t}+1} - t$. The study explores the long-term behavior of the distributions of $X_t$ and $Y_t$.

Renewal theorems in a periodic environment

Abstract

We study a renewal problem within a periodic environment, departing from the classical renewal theory by relaxing the assumption of independent and identically distributed inter-arrival times. Instead, the conditional distribution of the next arrival time, given the current one, is governed by a periodic kernel, denoted as . The periodicity property of is expressed as , where . For a fixed time , we define as the count of events occurring up to time . The focus is on two temporal aspects: , the time elapsed since the last event, and , the time until the next event occurs, given by and . The study explores the long-term behavior of the distributions of and .
Paper Structure (10 sections, 21 theorems, 84 equations)

This paper contains 10 sections, 21 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Remarks on the asymptotic distributions
  4. The Markov chain of the phases
  5. Backward recurrence time
  6. The canonical process and Renewal equations
  7. Proof of Theorem \ref{['th:backward']}
  8. Forward recurrence time
  9. Invariant measure of the sampled chain
  10. Application of the Harris's ergodic theorem

Key Result

Proposition 2.1

There exists a unique function $\rho \in L^1(\mathbb{T})$ satisfying eq:global-rho for all $t \in \mathbb{R}$. Moreover, it holds that $\rho \in C(\mathbb{T})$ and for all $t$, $\lambda_{\min} \leq \rho(t) \leq \lambda_{\max}$.

Theorems & Definitions (39)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • ...and 29 more