Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario ... Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA.
Statistics & Probability North-Holland
ON
September
Letters 8 (1989) 301-306
BEST
POSSIBLE
APPROXIMATIONS
OF LOCAL
1989
TIME
Miklos CSGRGG Department
of Mathematics
and Statistics,
Carleton University,
Ottawa, Ontario KIS 586, Canada
Lajos HORVATH Department
of Mathematics,
University of Utah, Salt Lake City, UT 84112, USA
Received July 1988 Revised August 1988
Abstract:
We establish
Keywords:
random
best possible
walks, Brownian
rates for approximating motion
(Wiener
random
process),
walk local time by Brownian
local time, strong
approximations
local time. (invariance
principles).
1. Results be independent identically distributed random variables (i.i.d. r.v.‘s) with P{ X, = l} = Let Xi, X,,..., -1) =i, and let S,=O, S,=X, + ... +X, (k=l, 2,...). Denote by t(n, n) (x= 0, f 1, f 2,. . . ; n = 1, 2,. . . ) the number of visits of the sequence S,, S,, . . . , S,, at x, i.e., P{X,=
t(x,
n)=#{k:
lO, lIV(t)I =l}, 7,=inf{t: f>7,_,, see that X, = W(7,) - W(T,_,) (i > 1) are i.i.d. r.v.‘s with P{ X, = l} = P{ X, = -l} = $. Also, T, - T,_, (i >, 1) are i.i.d. r.v.‘s with E(G-, - T,_~) = 1 and E(T, - T,-,)~ < cc. We put (I 2 = E( ,rl - 1)2.
(2.1)
Let a,(x) = L(x, Q,)+,) - L(x, T~(,)-~ ) (i > l), where ~(1) = min{k: k > 0, W(T~) = S, = x}, v(n) = min{k: k>v(n-l), W(T~)=S~=X} ( n > 2). Knight (1981, Lemma 5.1.13) showed that for any fixed x=0, 51, f2 ,..., Ea,(x)=l and y’=E(a,(x)-1)2, P(maxa,(x) x The approximation
> .Y>G (12/(no)“*)
exp(-u/16).
of (1.6) is based on the observation
(2.2) that
5Cx.n)
p
c
ai(x)-L(x,
7,)
