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 $$ J_{m,\bm\alpha}(B_H)(t) =\int_0^ts_m^{-\alpha_m}\int_0^{s_m}\cdots s_2^{-\alpha_2}\int_0^{s_2}s_1^{-\alpha_1}B_H(s_1)d s_1\; ds_2\cdots d s_m, $$ where $\alpha_1+\cdots+\alpha_i<H+i$, $i=1,\ldots,m$, $\bm\alpha=\bm\alpha_m=(\alpha_1,\ldots,\alpha_m)$. We show that \begin{align*} \liminf_{T\to \infty} \frac{(\log\log T)^{H+m}}{T^{H+m-\alpha}}\sup_{0\le t\le T}\left|\frac{ J_{m,\bm\alpha}(B_H)(t)}{t^{\alpha-\alpha_1-\cdots-\alpha_m}}\right| = a_H\left( \frac{\kappa_{H+m}}{1-\alpha/(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, \bm\alpha_{m-1}}(B_H)(s)}{s^{H+m-\alpha_1-\cdots-\alpha_{m-1}}}ds\right| &= \frac{\pi}{2}\frac{\sqrt{\beta(2H,1-H)}}{\prod_{i=1}^{m-1}\big(H+i-\alpha_1-\cdots-\alpha_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 \(J_{m, \bm\alpha}(B_H)\) 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.