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

2 ResultsNotation In what follows the phrase "S is an a.s.r.w.", (asymptotically stablerandom walk) will have the following meaning. S = (S n ; n 0) is a 1-dimensional random walk with S 0 = 0 and S n =P n1 X r for n 1 where X 1 ; X 2 ; are i.i.d. with F (x) = P (X 1 x) : S is either non-lattice, or it takes values on the integers and is aperiodic: there is a monotone increasing continuous function c(t) such that theprocess de…ned by X (n)t = S [nt] =c n converges weakly as n ! 1 to astable process Y = (Y t ; t 0): the process Y has index 2 (0; 2]; and := P (Y 1 > 0) 2 (0; 1):Remark 1 The case = 1; 2 (1; 2] is the spectrally negative case, and wewill sometimes need to treat this case separately. If < 1 then < 2 and theLévy measure of Y has a density equal to c + x 1 on (0; 1) with c + > 0;and then we can also assume that the norming sequence satis…esBut if = 1 we will havenF (c n ) ! 1 as n ! 1: (7)nF (c n ) ! 0 as n ! 1: (8)Here are our main results, where we recall that h y () denotes the densityfunction of the passage time over level y > 0 of the process Y: We will alsoadopt the convention that both x and are restricted to the integers in thelattice case.Theorem 2 Assume that S is an asrw. Then(A) uniformly for x such that x=c n ! 0;P (T x = n) v U(x)P (T = n) v U(x)n 1 L(n) as n ! 1 : (9)(B) uniformly in x n := x=c(n) 2 [D 1 ; D]; for any D > 1;If, in addition, < 1; andP (T x = n) v n 1 h xn (1) as n ! 1: (10)f x := P (S 1 2 [x; x + )) is regularly varying as x ! 1; (11)then(C) uniformly for x such that x=c n ! 1;P (T x = n) v F (x) as n ! 1: (12)3