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

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5.3 Proof when x=c n ! 1Proof. This time we write P (T x = n + 1) = P 41 P (i) ; where P (i) =P fA (i) \ (T x = n + 1)g andA (1) = (S n x); A (2) = (x < S n x Kc n );A (3) = (x Kc n < S n x c n ); and A (4) = (S n > x c n ):We note …rst that for 2 (0; 1);P (1)lim supn!1 F (x)P (1)lim infn!1 F (x)P (S n x)F ((1lim supn!1 F (x))x)= (1 ) ;P (max rn S r x; jS n j x)F ((1 + )x)lim infn!1F (x)= (1 + ) : (49)Next, using (27),P (2)lim supn!1 F (x)P (S n > x)F (Kc n )lim supn!1 F (x)nF (x)F (c n )lim sup= Cn!1 F (x)K (K) : (50)To deal with the next term, we use (27) again to see that for any …xed K;P (x Kc n < S n x) is uniformly o(nF (x)): Since P (3) F (c n )P (x Kc n n) vnP ( > n)x 1 F (x); so we can assume thatxg(n; x)P ( > n)sup< 1: (52)n>0; x>Kc n F (x)17
Then for large enough n and x > Kc nP (4) ===nX0bcX ncy=0bc ncCF (x)xP ( > n)CF (x)xP ( > n)CF (x)xP ( > n)CF (x)xP ( > n)Xz=0nX0ybcX ncy=0bcX ncz=0bcX ncz=0g(n m; x y)g (m; z)F (y + z)bcX ncy=0bc ncXz=0bc ncXy=0bcX ncw=zbc ncXz=0yzyg (m; z)F (y + z)v(z)F (y + z)v(z)F (y + z)v(z)F (w):Now a summation by parts and the fact that V F 2 RV (y ! 1) shows that asyX yXyXv(z)F (w) v V (z)F (z) v yV (y)F (y)=(1z=0 w=zz=0):So for all large enough n we have the boundP (4) CF (x)1 c n V (c n )nxP ( > n)CF (x) 1 c nnxP ( > n)P ( > n) v C1 c n F (x) (53)xThe result follows from (49)-(53) and appropriate choice of ; K; and :Remark 16 The assumption (11) is not strictly necessary for (52) to hold,and this is the only point where we use this assumption. In fact, if the followingslightly weaker version of (52),c n g(n; x)P ( > n)sup< 1; (54)n>0; x>Kc n F (x)were to hold, then (53) would hold with c n replacing x in the denominator, andthe proof would still be valid, by choosing small.18
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 16 and 17: so one part of (45) will follow if
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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