Stationary distribution

1
Stationary distribution

• is a stationary distribution for Pt if ?? ?Pt
  for all t.
• Theorem
• is stationary for Pt 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 Pt as .
• Death process case

3
Persistence

• A chain is irreducible if Pij(t) gt 0 for some t
  and for all pairs i,j in S.
• Fact (Lévy dichotomy) Either Pij(t) gt 0 for all
  t, or Pij(t) 0 for all t.
• We call a state persistent if
• Let Yn 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

• j is persistent iff
• pii(t)gt0 for all t
• Proof
• For the discrete skeleton j is persistent iff
  
• i.e. iff
• (ii) so
• For any t pick n so large that thn. 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?