Extreme values of Markov population processes

In contrast to the classical theory of partial sums of independent and identically distributed random variables, the maximum value taken by a component of a Markov population process xN is typically largely determined by the variation in its mean, rather than by stochastic fluctuation. A closer approximation to its distribution is found by considering the supremum of V(t) - N c(t) for a suitable centred Gaussian process V, where c incorporates the effect of the variation in the mean of xN. Under appropriate conditions, it is shown that this has a distribution which is normally distributed, to within an error of order N- log N, and expressions for the mean and variance of the approximating distribution are derived.