Large Deviation Rates for Branching Processes--I. Single Type Case

Abstract

Let {Z(n)}(0)(infinity) be a Galton-Watson branching process with offspring distribution {p(j)}(0)(infinity). We assume throughout that p(0) = 0, pj = 0, p(j) not equal 1 for any j >= 1 and 1 < m = Sigma jp(j) infinity of P( vertical bar Z(n+1)/z(n) - m vertical bar > epsilon), P(vertical bar W-n - W vertical bar > epsilon), P( vertical bar Z(n+1)/z(n) - m vertical bar > epsilon vertical bar W >= a) for epsilon > 0 and a > 0 under various moment conditions on {p(j)}. It is shown that the rate for the first oue is geometric if p(1) > 0 and supergeometric p(1) = 0, while rartes for the other two are always supergemetric under a finite moment generating functions hypothesis.