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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)].

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