Matrix Properties of a Class of Birth-Death Chains and Processes

When
Start: 04/24/2017 - 4:15pm
End  : 04/24/2017 - 5:15pm

Category
Applied Math Seminar

Speaker
Alan Krinik (Cal Poly Pomona)

Abstract

We consider various recurrent birth-death chains on state space S1 = {0,1,2,...,H} and its associated dual birth-death chain on state space S2 = {−1,0,1,2,...,H} having absorbing states −1 and H. Assume P and P∗ are the one-step transition probability matrices of each birth-death chain respectively.

Conclusions:

1. P and P∗ have the same set of eigenvalues.

2. An explicit, simple formula for the eigenvalues of P (and P∗) is described as a function of H.

3. Pn and (P∗)n can be expressed exactly for n ∈ N.

Conclusion 2 follows from some nice linear algebra results on certain types of tridiagonal matrices found in Kouachi (2006, 2008). Our conclusion 3 has implications for finite-time gambler’s ruin problems. Some of the preceding results extend beyond birth-death chains to certain Markov chains. Our results for stochastic matrices suggest further ways to generalize Kouachi’s work. Explicit formula for finding transition probability functions of certain birth-death processes is also described.

Where
Emmy Noether Rm Millikan 1021 Pomona College