Conditioned Random Walks and Lévy Processes

Let X1; X2; ::: be independent, identically distributed, zero mean random variables with ( )-regularly varying tails, > 1. For Sn = Pn i=1Xi, it is known that under these distributional assumptions, P(Sn > x) nP(X1 > x) as x ! 1, uniformly for x cn for any constant c > 0. Here, we show that the process Mn = maxfSi i : i ng, for any constant 0, behaves in a similar manner. This allows us to generalise Durrett’s results [7], by showing that, without any further assumptions, n S[nt]; 0 t 1jSn > na and n S[nt]; 0 t 1jMn > na for any constant a > 0, both converge weakly to a simple process consisting of a single ‘large jump’. We show that similar results for general Lévy processes, extending the work of Konstantopoulos and Richardson [10], who dealt with the special case of spectrally positive processes. 1 Introduction Let X1; X2; ::: be independent, identically distributed, zero mean random variables with distribution function F; and de…ne a random walk by Sn := Pn k=1Xi, n 1; and S0 = 0: We will be interested in the case that F (x) := P(Xi > x) x L(x) as x!1; (1) where > 1, and L denotes a slowly varying function. Note that applying Nagaev’s result [11] to the random variables Xi yields P(Sn > x) u nP(X1 > x); (2) where we will from now on use u to denote that the asymptotic relation is ‘uniform, as n!1 for x nc and any constant c > 0’. Nagaev’s result implies that the ‘one large jump’principle holds. Durrett [7] showed, in the special case that X1 has …nite variance, so that automatically the in (1) is at least 2, that if we condition on the event Sn > na; where a > 0 is …xed, there is a functional limit theorem for (n S[nt]; 0 t 1): Since (2) holds for any > 1 under the sole additional assumption that EX1 = 0; one of our aims is to extend Durrett’s result to this case. Another is to show, again in accord with the "one large jump principle", that the analogue of (2)