# Professor David McDonald University of Ottawa

## Renewal Theory and the Local Limit Theorem

The classical renewal and local limit theorems for discrete, independent random variables can be proved for nonidentically distributed random variables. The key idea is to find embedded Bernoulli random variables hidden in each of the summands! Since sums of Bernoulli random variables are well understood we can obtain the ``smoothness'' of the distribution of the sum which is at the heart of the proofs of both the renewal and local limit theorems.

This lecture is accessible to undergraduate students who have had a first course in probability.

### References

McDonald, D. (1978). On semi-Markov and semi-Markov and semi-regenerative processes I, Zeitschrift f r Wahrscheinlichkeitstheorie Verw. Gebiete, 42, 261-277.

McDonald, D. (1978). On semi-Markov and semi-regenerative processes II, Annals of Probability, Vol.6, No.6, 995-1014.

McDonald, D. (1979). A local limit theorem for large deviations of sums of independent, non-identically distributed random variables. Ann. Probability, Vol.7, 526-531.

McDonald, D. (1979). On local limit theorems for integer valued random variables. Teor. vernoiatnostei i ee Primen, Vol.24, No.3, 607-614.

Davis, B., McDonald, D. (1995). An Elementary Proof of the Local Central Limit Theorem. Journal of Theoretical Probability Theory. Vol. 8, No. 3, 693-701.

Refreshments will be served in Skiles 239 prior to the lecture.

## Rare events for a Markov chain

The performance of a telecommunication system is often measured by the frequency or probability of rare events such as a buffer overflow. Such a system is modeled by a Markov chain with a large state space and stationary distribution p. A rare event corresponds to an excursion to a forbidden set F with small stationary probability p(F).

The distribution of the duration of such excursions is asymptotically exponential and during each excursion there is a small clump of visits to the forbidden set. The Poisson clumping heuristic due to Aldous connects the mean clump size and the steady state probability of the forbidden set with the mean excursion length.

Here we consider one node in a queueing network whose transitions may be represented by an Markov additive process which recurs to a reflection point at zero. The steady state p is not known so the clumping heuristic doesn't work. Nevertheless it is possible to perform an h-transformation or {twist} which turns the additive process around much like the one dimensional associated random walk described in Feller Volume II. This twist automatically yields the rough asymptotics of the mean excursion length. The exact asymptotics are given by probabilistic expression involving the twisted Markov additive process. The steady state of the network conditioned on the specified node being in F also follows. The twisted process is transient to F so a computer simulation of the rare event is very quick. The hitting distribution of the twisted process on F immediately yields that of the original Markov chain.