Approximation of stopped Brownian local time by diadic crossing chains

Let B(t) be a Brownian motion on R, B(0) = 0, and for [alpha]n:= 2-n let Tn0 = 0, Tnk+1 = inf{t> Tnk:B(t)-B(Tnk) = [alpha]n}, 0 [less-than-or-equals, slant] k. Then B(Tnk):= Rn(k[alpha]2n) is the nth approximating random walk. Define Mn by TnMn = T(-1) (the passage time to -1) and let L(x) be the local time of B at T(-1). The paper is concerned with 1. (a) the conditional law of L given [sigma](Rn), and 2. (b) the estimator E(L(·)[sigma](Rn)). Let Nn(k) denote the number of upcrossings by Rn of (k[alpha]n, (k + 1)[alpha]n) by step Mn. Explicit formulae for (a) and (b) are obtained in terms of Nn. More generally, for T = TnKn, 0[less-than-or-equals, slant]Kn [set membership, variant] [sigma](Rn), let L(x) be the local time at T, and let N±n(k) be the respective numbers of upcrossings (downcrossings) by step Kn. Simple expressions for (a) and (b) are given in terms of N±n. For fixed measure [mu] on R, 2nE[[integral operator](E(L(x)[sigma](Rn)) - L(x))2[mu](dx)[sigma](Rn] is obtained, and when [mu](dx) = dx it reduces to . With T kept fixed as n --> [infinity], this converges P-a.s. to .