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

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so one part of (45) will follow if we can show thatObviously, for any B > 0P (T x > n) CU(x)P ( > n): (46)P (T x > n) = X y0P (S n = x y; T x > n)X0yBc nP (S n = x y; T x > n) + P (S n < x Bc n ; T x > n):Since x=c n ! 0; we can use the invariance principle in Proposition 9 to …x Blarge enough to ensure that P (S n < x Bc n jT x > n) 1=2 for all su¢ cientlylarge n: We can also use Corollary 13 to getXP (S n = x y; T x > n) CU(x) XV (y)nc n0yBc n 0yBc nand thus (46) is established. = 1:) Now (42) giveslim lim sup"#0 n!1 CU(x)Bc nV (Bc n ) CU(x)nc n nP ( > n) CU(x)P ( > n);1U(x)P ( = n)(Note that this proof is also valid for the caseXy2[0;"c n]2f(0)lim lim sup"#0 n!1 nc n P ( = n)P (S n > xXy2[0;"c n]y; T x > n)F (y)F (y)V (y);and since < 1; (105) of [21] shows that this is zero.In the case = 1; a di¤erent proof is required. We make use of theobservation in [21] that there is a sequence n # 0 with n c n ! 1 and suchthat nF ( n c n ) ! 0: This givesXP (S n = x y; T x > n)F (y) P (T x > n)F ( n c n )y> nc n= o(U(x)P ( > n)=n) = o(U(x)P ( = n));where we have used (46). Using (23) of Proposition 12 givesXy nc nP (S n = xy; T x > n)F (y) v f(0)U(x)!(n)nc nas n ! 1;uniformly in x, where !(n) = P 0y nc nV (y)F (y): But in [21], it is shownthat nc n P ( = n) v f(0)!(n), so we deduce from (41) that P (T x = n + 1) vU(x)P ( = n); as required.15
5.2 Proof when x=c n = O(1)Proof. Again we treat the case < 1 …rst; and start by noting that for anyB > 0;XP (S n = x y; T x > n)F (y) F (Bc n ) v 1 as n ! 1;nB y>Bc nuniformly in x 0: Also by Corollary 13, for b > 0,Xybc nP (S n = xy; T x > n)F (y) CU(x) P ybc nV (y)F (y)nc n: (47)Since V F 2 RV ( ); P yzV (y)F (y) v zV (z)F (z)=(1 ); and we see thatwhen x Dc n ;U(x) P ybc nV (y)F (y)c nv b 1 U(x)V (c n )F (c n ) Cb 1 D U(c n )V (c n )=n Cb 1 D ;where we have used (15) from Corollary 8. We conclude the proof by showingthatlim lim n XP (S n = x y; T x > n)F (y) = C; (48)b#0; B"1 n!1 h xn (1)bc n n)F (y) nF ( n c n ) ! 0;y> nc nandnX0y nc nP (S n = xv nP ( > n)p(x n)c nvXy; T x > n)F (y)0y nc nV (y)F (y)np(x n)!(n)v p(x n)P ( = n)v p(x n );P ( > n)nc n P ( > n)and the result follows from the identity (33) in Proposition 15, since again therewould be a contradiction if k 9 6= 1.16
Page 1: Local Behaviour of First Passage Pr
Page 5: From this we get immediately a stre
Page 8 and 9: From this we can deduce the followi
Page 10 and 11: in t; of the LHS of (34) is the sam
Page 12 and 13: where lim n; () = 1 is shorthand fo
Page 14 and 15: (iii) In this case it is P 2 that d
Page 18 and 19: 5.3 Proof when x=c n ! 1Proof. This
Page 20 and 21: 6 The non-lattice caseWe indicate h
Page 22 and 23: An asymptotic lower bound is given
Page 24 and 25: References[1] V. I. Afanasyev, C. B

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