Asymptotic Results for Spectrally Positive Compound Poisson Processes

Abstract

Finite excursions away from zero of a spectrally positive compound Poisson process with a negative drift can always be decomposed into two parts lying above and below zero, respectively. This paper is concerned with the asymptotic relationships among the lengths and heights of these two parts. Our results state that both their lengths and heights are asymptotically strongly dependent and exhibit a scale symmetry.

Figures (2)

• Figure 1.1: A typical excursion away from $0$ that can be decomposed into two parts lying above and below zero.
• Figure 3.1: Partition of $\mathbb{R}_{+}^2$ into seven regions. Here $A_1=\{0<x<\infty, 0<y<\delta ah\}$, $A_2=\{0<x<\delta h, y>\delta ah\}$, $A_3=\{x>h, y>ah\}$, $A_4=\{\delta h<x<h, y>ah\}$, $A_5=\{x>h, \delta ah<y<ah\}$, $A_6=\{\delta h<x<h, (1-\delta)ah<y< ah\}$, $A_7=\{\delta h<x<h, \delta ah<y<(1-\delta) ah\}$.

Theorems & Definitions (10)

• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Lemma 2.1
• Corollary 2.2
• Lemma 3.1
• Lemma 3.2