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

to that in (i) to show thatP 1 =bncX0xXz=0~p (y)e(n(1 ))~p(y=c n r )g(r; x z)(1 + o(1))e(n r)!xXu(xz=0= U(x)~p (y)(1 + o(1));e(n(1 ))z)(1 + o(1))where ~p (y) = sup 0rn ~p(y=c n r ) = ~p(y n ) + "(n; ) and lim n; "(n; ) =0: In the same way we get P 1 U(x)~p (y)=e(n)(1 + o(1)), where ~p (y) =inf 0rn ~p(y=c n r ); and we deduce thate(n)P 1lim= 1: (38)n; ~p(y n )U(x)Since inff~p(y) : y 2 [D 1 ; D] > 0; the result will follow if we can show that forany …xed > 0e(n)(P 2 + P 3 )lim sup= 0: (39)n!1 U(x)However (37) still holds, but note now thatXy^xU(x 1 z)v(y z) =z=0xXU(x 1 z)v(y z)z=0 U(x)(V (y) V (y x 1)) = o(U(x)V (y)):and since the analogue of (14) holds, viz V (c n ) v k 10 =P ( > n); we see that e(n)P 3 U(x)V (y) e(n)= oU(x)d(n)U(x)1= o(nP ( > n)P ( > n) ) = o(1):Finallye(n)P 2 = e(n) X xXg(r; x z)g (n r; y z)A 2z=0 f(0)e(n) X A 2xXz=0U(x z)~p((y z)=c n r )d(r)e(n r)Ce(n)nxU(x)d(bnc)e(bnc) = O(xU(x) ) = o(U(x)):Thus (39) is established, and the result (24) follows. Since (25) is (24) for Swith x and y interchanged, modi…ed to take account of the di¤erence betweenstrict and weak ladder epochs, we omit it’s proof.c n;12