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On the character of convergence to Brownian local time. (English. Russian original) Zbl 0534.60032
Sov. Math., Dokl. 27, 400-405 (1983); translation from Dokl. Akad. Nauk SSSR 269, 784-788 (1983).
The local time, l(t,x), of a Brownian motion process w(s) \((E(w^ 2(s))=Ds\), \(0<s<\infty)\) is the density of the occupation measure \(\mu_ t(A)=\int^{t}_{0}l_ A(w(s))ds,\) where \(l_ A\) is the indicator function of A, A any Borel subset of R. The author provides several results involving convergence to local time.
In particular, it is shown that the finite-dimensional distributions of various processes \(L^ i_{\epsilon}(t,x)\) converge, as \(\epsilon\to 0\), to the finite-dimensional distributions of \(d_ ir(t,x)\), where \(d_ i\) is a constant depending upon i and \(r(t,x)=w_ x(l(t,x))\), where \(w_ x(s)\), \(s\geq 0\), is a family of standard Brownian motion processes independent of w(s) and of each other for differing \(x_ s\). For example, for \(L_{\epsilon}=(l(t,x+\epsilon)-1(t,x))/e^{1/2},\) the appropriate constant is d-2/\(\sqrt{D}\). The key idea is to view w(s) as the limiting process of normalized random walks and then investigate convergence of characteristics of the random walks to the local time.
Reviewer: E.Boylan

60F17 Functional limit theorems; invariance principles
60J55 Local time and additive functionals
60J65 Brownian motion
60G50 Sums of independent random variables; random walks