Page 238 - Applied Probability

10. Molecular Phylogeny
224
Using this, compute the 2 × 2 matrix exponential

α
exp s
β
and find its limit as s →∞. −α −β ,
5. Consider a continuous-time Markov chain with infinitesimal transi-
tion matrix Λ = (Λ ij ) and equilibrium distribution π. If the chain is
at equilibrium at time 0, then show that it experiences t π i λ i tran-
sitions on average during the time interval [0,t], where λ i = i Λ ij .
j =i
6. Let Λ be the infinitesimal transition matrix and π the equilibrium
distribution of a reversible Markov chain with n states. Define an
inner product u, v π on complex column vectors u and v with n
components by

= u i π i v ,
∗
u, v π i
i
where ∗ denotes complex conjugate. Verify that Λ satisfies the self-
adjointness condition
Λu, v π = u, Λv π .
Conclude by standard arguments that Λ has only real eigenvalues.
7. Let Λ = (Λ ij )be an m × m matrix and π =(π i )bea 1 × m row
vector. Show that the equality π i Λ ij = π j Λ ji is true for all pairs
t
(i, j) if and only if diag(π)Λ=Λ diag(π), where diag(π) is a di-
agonal matrix with ith diagonal entry π i . Now suppose Λ is an in-
finitesimal generator with equilibrium distribution π.If P(t)= e tΛ
is its finite-time transition matrix, then show that detailed balance
π i Λ ij = π j Λ ji for all pairs (i, j) is equivalent to finite-time detailed
balance π i p ij (t)= π j p ji (t) for all pairs (i, j) and times t ≥ 0.
8. Let Λ be the infinitesimal transition matrix of a Markov chain, and
1
suppose µ ≥ max i λ i .If R = I + Λ, prove that R has nonnegative
µ
entries and that
∞ i
−µt (µt) i
S(t) = e R
i!
i=0
coincides with P(t). (Hint: Verify that S(t) satisfies the same defining
differential equation and the same initial condition as P(t).)

9. Let P(t)=[p ij (t)] be the finite-time transition matrix of a finite-state
irreducible Markov chain. Show that p ij (t) > 0 for all i, j, and t> 0.
Thus, no state in a continuous-time chain displays periodic behavior.
(Hint: Use Problem 8.)