Download presentation
Presentation is loading. Please wait.
1
Stationary distribution is a stationary distribution for P t if = P t for all t. Theorem: is stationary for P t iff G = 0 (under suitable regularity conditions). Proof:
2
Death process As for discrete time, the stationary distribution can be thought of as the limit of P t as. Death process case:
3
Persistence A chain is irreducible if P ij (t) > 0 for some t and for all pairs i,j in S. Fact (Lévy dichotomy): Either P ij (t) > 0 for all t, or P ij (t) = 0 for all t. We call a state persistent if Let Y n = X(nh) be the discrete skeleton of X. Let Q be the transition matrix for Y so persistence in continuous time is the same as persistence for the discrete skeleton
4
Some further facts (i)j is persistent iff (ii)p ii (t)>0 for all t Proof: (i)For the discrete skeleton j is persistent iff i.e. iff (ii)so For any t pick n so large that t≤hn. By C-K
5
Birth, death, immigration and emigration Let The n are called death rates, and the n are called birth rates. The process is a birth and death process. If n = n + we have linear birth with immigration. If n = ( + )n we have linear death and emigration.
6
Generator Stationary distribution G = 0 yields whence When is this a stationary distribution?
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.