The behavior of the ratio of a small-noise Markov chain to its deterministic approximation
For each N≧1, let {XN(t, x), t≧0} be a discrete-time stochastic process with XN (0) = x. Let FN (y) = E(XN (t + 1) | XN (t) = y), and define YN (t, x) = FN(YN(t – 1, x)), t≧1 and YN (0, x) = x. Assume that in a neighborhood of the origin FN (y) = mNy(l + O(y)) where mN > 1, and define for δ> 0 and x> 0, υ N (δ, x) = inf{t:xmt N >δ}. Conditions are given under which, for θ> 0 and ε> 0, there exist constants δ > 0 and L <∞, depending on εand 0, such that This result together with a result of Kurtz (1970), (1971) shows that, under appropriate conditions, the time needed for the stochastic process {XN (t, 1/N), t≧0} to escape a δ -neighborhood of the origin is of order log Νδ /log mN . To illustrate the results the Wright-Fisher model with selection is considered.