Some Singular Sample Path Properties of a Multiparameter Fractional Brownian Motion

Abstract

We obtain a spectral representation and compute the small ball probabilities for a (non-increment stationary) multiparameter extension of the fractional Brownian motion. We derive from these results a Chung-type law of the iterated logarithm at the origin and exhibit the singular behaviour of this multiparameter fractional Brownian motion, as it behaves very differently at the origin and away from the axes. A functional version of this Chung-type law is also provided.

Appendix: Proof of Lemma 4.1

For the original proof, see Lemma 5.3 of [7]. We make here the necessary modifications.

$$\begin{aligned} \left( \log \log r^{-1}\right) ^{h/\nu +1/2} \Vert \eta ^{(\ell )}_r - f\Vert _\infty&= \frac{(\log \log r^{-1})^{h/\nu }}{r^{\nu h}} \left\| \varvec{B}(r\cdot ) - r^{\nu h}\sqrt{\log \log r^{-1}} f\right\| _\infty \\&\ge \frac{(\log \log r^{-1})^{h/\nu }}{r^{\nu h}} \left\| \varvec{B}(s\cdot ) - r^{\nu h}\sqrt{\log \log r^{-1}} f\left( \frac{s}{r}\cdot \right) \right\| _\infty \\&\ge \frac{(\log \log u^{-1})^{h/\nu }}{u^{\nu h}} \left\| \varvec{B}(s\cdot ) - r^{\nu h}\sqrt{\log \log r^{-1}} f\left( \frac{s}{r}\cdot \right) \right\| _\infty . \end{aligned}$$

Now choosing $a= s^{\nu h}\sqrt{\log \log s^{-1}}$ and $b=u^{\nu h}\sqrt{\log \log u^{-1}}$,

$$\begin{aligned} \left\| \varvec{B}(s\cdot ) - r^{\nu h}\sqrt{\log \log r^{-1}} f\left( \frac{s}{r}\cdot \right) \right\| _\infty\ge & {} \left\| \varvec{B}(s\cdot ) - a f\right\| _\infty - b\left\| f-f\left( \frac{s}{r}\cdot \right) \right\| _\infty \\&- (b-a)\Vert f\Vert _\infty \end{aligned}$$

and we find a bound for each of the last two terms (the first one is exactly the one given in the Lemma). We need the following inequality for $f\in H^\nu ,\,s,t\in [0,1]^\nu$:

$$\begin{aligned} |f(s)-f(t)|^2 \le M_1 \Vert s-t\Vert ^{2h} \Vert f\Vert _{\nu }^2, \end{aligned}$$

which follows from approximation of f by linear combinations of simple functions of the form $\lambda (\mathbf {1}_{[0,t_i]}\mathbf {1}_{[0,\cdot ]})$ and the upper bound in Lemma 2.6 (where the constant $M_1$ comes from). Thus,

$$\begin{aligned} - b \frac{(\log \log u^{-1})^{h/\nu }}{u^{\nu h}} \left\| f-f\left( \frac{s}{r}\cdot \right) \right\| _\infty \ge -M_1 \left( \log \log u^{-1}\right) ^{h/\nu +1/2} \left( 1-\frac{s}{u}\right) ^h \Vert f\Vert _{\nu }. \end{aligned}$$

For the last term, we use the fact that:

$$\begin{aligned} b-a \le \sqrt{u^{2\nu h} \log \log u^{-1} - s^{2\nu h}\log \log s^{-1}}, \end{aligned}$$

which ends the proof of this lemma.