Local Behaviour of First Passage Probabilities - MIMS - The ...

(ii) For any D > 1; uniformly for y n 2 [D 1 ; D];P (S n 2 (x y ; x y]; T x > n) v U(x)P ( > n)~p(y n)c nas n ! 1 and x n ! 0;and uniformly for x n 2 [D 1 ; D];P (S n 2 (x y ; x y]; T x > n) v V (y)P ( > n)p(x n)c nas n ! 1 and y n ! 0:(iii) For any D > 1; uniformly for x n 2 [D 1 ; D] and y n 2 [D 1 ; D];(60)(61)P (S n 2 (x y ; x y]; T x > n) v q x n(y n )as n ! 1: (62)c nOnce we have these results, we deduce Theorem 2 by applying them to amodi…ed version of (57), viz1XP (S n 2 (x (n + 1); x n]; T x > n)F (n) P (T x = n + 1)01XP (S n 2 (x (n + 1); x n]; T x > n)F ((n + 1));0and letting ! 0: So the key step is establishing Proposition 19, and weillustrate how this can be done by proving (59).Proof. We want to apply Lemma 17 to (58), but technically the problem isthat we can’t do this directly, as we did in the lattice case. The …rst step is toget an integrated form of (58), and it is useful to separate o¤ the term r = 0 ;so that for x; y 0;P (S n 2 (x y ; x y]; T x > n) = G (n; [fy xg + ; y + x))1 fxy+g + P ~ ;(63)wherenXZ Z~P =G(r; x dz)G (n r; dw z);==r=1nXZr=1nXZr=1yw