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On some limit theorems for occupation times of one dimensional Brownian motion and its continuous additive functionals locally of zero energy. (English) Zbl 0618.60080
Let $$B_ t$$ be a standard Brownian motion and f be a continuous function. The author investigates the convergence in law on the space of continuous functions for the family of stochastic processes $$\xi_{\lambda}(t)=a_{\lambda}\int^{\lambda t}_{0}f(B_ s)ds$$ as $$\lambda\to \infty$$. The limit process depends on the nonrandom norming function $$a_{\lambda}$$ and assumptions on f($$\cdot)$$. The known results are connected with the case when f has a compact support and $$I=\int f(x)dx\neq 0$$ or $$I=0$$ but $$<f,f>=2\int^{\infty}_{- \infty}(\int^{x}_{-\infty}f(u)du)^ 2dx<\infty.$$
In this article the case when $$f\not\in L^ 1({\mathbb{R}})$$ or the case when $$I=0$$ but $$<f,f>=\infty$$ are discussed. For this purpose some classes of continuous additive functionals locally of zero energy are introduced: class $$C^ a_ t$$ of functionals corresponding to Cauchy’s principle value for 1/(x-a), class $$H^ a(-1-\alpha,t)$$ corresponding to Hadamard’s finite part (f.p.) $$(x-a)_+^{-1-\alpha}$$ and class $$H^ a(1+\beta,t)$$ corresponding to f.p. $$(x-a)_+^{\beta -1}.$$
Under some additional assumptions on f($$\cdot)$$ the limit process for the $$\xi_{\lambda}$$ can be expressed with the help of this classes of functionals.
Reviewer: N.M.Zinchenko

##### MSC:
 60J65 Brownian motion 60J55 Local time and additive functionals 60F17 Functional limit theorems; invariance principles
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