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On a functional limit result for increments of a fractional Brownian motion. (English) Zbl 1001.60033
Let $$X(t)$$, $$t\geq 0$$, be a standard $$\alpha$$-fractional Brownian motion with $$X(0)=0$$, $$0<\alpha<1$$, and let $$K$$ be the unit ball of RKHS of the measure induced by $$X$$ on $$C_0[0,1]$$. For $$0<h<1$$ and $$0\leq t\leq 1-h$$ consider the process $M_{t,h}(x)=(X(t+hx)-X(t))/\sqrt{2h^{2\alpha}\log h^{-1}},\quad 0\leq x\leq 1,$ of small increments of $$X(t)$$. It is proved that a.s. as $$h\to 0$$: $\sup_{0\leq t\leq 1-h}\inf_{f\in K}\|M_{t,h}-f\|_\infty \to 0$ and for any $$f\in K$$, $\sup_{0\leq t\leq 1-h}\sup_{0\leq x\leq 1}(M_{t,h}(x)-f(x))\to 0$ as well as $\sup_{0\leq t\leq 1-h}\sup_{0\leq x\leq 1}(f(x)-M_{t,h}(x))\to 0.$ As a special case Lévy’s modulus of continuity for $$X$$ is obtained. For $$\alpha=\frac 12$$ these results were obtained by B. Chen [Ph.D. Thesis, Univ. Carleton of Canada (1998)]. The main tool is some large deviation result for Gaussian processes which is related to that given by V. Goodman and J. Kuelbs [Probab. Theory Relat. Fields 88, No. 1, 47-75 (1991; Zbl 0695.60040)].

##### MSC:
 60F15 Strong limit theorems 60J65 Brownian motion 60G15 Gaussian processes 60G17 Sample path properties
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