Tid: 6 november 2000 kl 1515-1700

Plats : Seminarierummet 3733, Institutionen för matematik, KTH, Lindstedts väg 25, plan 7. Karta!

Föredragshållare: Torkel Erhardsson, Matematisk statistik, KTH.

Titel: Compound Poisson approximation for visits to rare sets by certain stationary Markov chains and renewal reward processes.


Let $\eta$ be a stationary discrete time Markov chains which is ``strongly aperiodic Harris recurrent'' (e.g., an irreducible and positive recurrent Markov chain on a countable state space), with stationary distribution $\mu$. What can be said about the distribution of the number of visits up to time $\,n\,\,$ by $\eta$ to a subset $\,S_1$ of the state space such that $\mu(S_1)$ is small?

We will here give a bound for the total variation distance between this distribution and a compound Poisson distribution. It will be shown how the bound can be derived using Stein's method, regenerative properties of Harris recurrent Markov chains, and couplings. The bound depends only on much studied quantities like hitting probabilities and expected hitting times, which can be easily computed if the state space is finite. Under certain conditions the bound is of order close to $\log(n\mu(S_1))\mu(S_1)$, or even $\mu(S_1)$.

If time permits, it will also be indicated how these results can be extended, using point process theory, to compound Poisson approximation for the accumulated rewards of stationary renewal reward processes in discrete or continuous time.

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