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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

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