Chung-type laws of the iterated logarithm for $m$-fold weighted integrated fractional processes

Abstract

Let $\{B_H(t);t\ge 0\}$ be a fractional Brownian motion of order $H\in (0,1)$, and $J_{m,\alpha }(B_H)$ be the $m$-fold weighted integrals of $B_H$ defined as $$J_{m,\mathbf{\alpha }}(B_H)(t)={\int }_0^t{s}_m^{-{\alpha }_m}{\int }_0^{s_m}\cdots s_2^{-{\alpha }_2}{\int }_0^{s_2}{s}_1^{-{\alpha }_1}{B}_H(s_1)d{s}_{1}d{s}_2\cdots d{s}_m,$$ where ${\alpha }_1+\cdots +{\alpha }_i<H+i$, $i=1,\dots ,m$, $\mathbf{\alpha }={\mathbf{\alpha }}_m=({\alpha }_1,\dots ,{\alpha }_m)$. We show that \begin{align*} \liminf_{T\to \infty} \frac{(\log\log T)^{H+m}}{T^{H+m-α}}\sup_{0\le t\le T}\left|\frac{ J_{m,\bmα}(B_H)(t)}{t^{α-α_1-\cdots-α_m}}\right| = a_H\left( \frac{κ_{H+m}}{1-α/(H+m)}\right)^{H+m}\;\; a.s. \end{align*} for all $\alpha <H+m$, and \begin{align*} \liminf_{T\to \infty} & \sqrt{\frac{\log\log\log T}{\log T}} \sup_{1\le t\le T}\left|\int_1^t \frac{J_{m-1, \bmα_{m-1}}(B_H)(s)}{s^{H+m-α_1-\cdots-α_{m-1}}}ds\right| &= \fracπ{2}\frac{\sqrt{β(2H,1-H)}}{\prod_{i=1}^{m-1}\big(H+i-α_1-\cdots-α_i\big)}\;\; a.s., \end{align*} where $a_H$ is an explicit constant with $a_{\frac{1}{2}}=1$, ${\kappa }_{\lambda }$ is a constant which depends only on $\lambda$, and $\beta (a,b)$ is the beta function.In particular, the exact value of a Chung-type law of the iterated logarithm established by Duker, Li and Linde (2000) is found, and as an application, the Chung-type law of the iterated logarithm for the randomized play-the-winner rule is established. The small ball probabilities of \(J_{m, \bmα}(B_H)\) are established to show the liminf behaviors. Similar Chung-type laws of the iterated logarithm and small ball probabilities for a Riemann-Liouville fractional process are also established.

34 pages. The paper is submitted to Science in China-Mathematics