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

TL;DR

This paper establishes Chung-type laws of the iterated logarithm for m-fold weighted integrated fractional processes, providing exact liminf behaviors and small ball probabilities, with applications to the play-the-winner rule.

Contribution

It derives exact Chung-type laws of the iterated logarithm for complex fractional processes, extending previous results and including applications to stochastic decision rules.

Findings

Established exact liminf behaviors for weighted integrated fractional processes.

Derived small ball probabilities for these processes.

Applied results to the randomized play-the-winner rule.

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.