Let {BH(t); t ⩾ 0} be a fractional Brownian motion of order H ∈ (0, 1), and Jm,α(BH) be the m-fold weighted integral of BH defined as
\(J_{m,\alpha}(B_H)(t) = \int_{0}^{t} s_m^{-\alpha_m} \int_{0}^{s_m} \cdots s_2^{-\alpha_2} \int_{0}^{s_2} s_1^{-\alpha_1} B_H(s_1)\, ds_1\, ds_2 \cdots ds_m,\)
where α1 + ⋯ + αi < H + i, i = 1,…, m, and α = αm = (α1,…, αm). We show that
\(\liminf_{T \to \infty} \frac{(\log \log T)^{H+m}}{T^{H+m-\alpha}} \sup_{0 \leqslant t \leqslant T} \left| \frac{J_{m,\alpha}(B_H)(t)}{t^{\alpha - \alpha_1 - \cdots - \alpha_m}} \right| = a_H \left(\frac{\kappa_{H+m}}{1 - \alpha/(H + m)} \right)^{H+m} \quad \text{a.s.}\)
for all α < H + m, and
\(\liminf_{T \to \infty} \sqrt{\frac{\log \log \log T}{\log T}} \sup_{1 \leqslant t \leqslant T} \left| \int_{1}^{t} \frac{J_{m-1,\alpha_{m-1}}(B_H)(s)}{s^{H+m-\alpha_1-\cdots-\alpha_{m-1}}} \, ds \right|\\= \frac{\pi}{2} \frac{\sqrt{B(2H, 1-H)}}{\prod_{i=1}^{m-1} (H + i - \alpha_1 - \cdots - \alpha_i)} \quad \text{a.s.},\)
where aH is an explicit constant with \(\alpha_{\frac{1}{2}} = 1,\kappa_{\lambda}\) is a constant which depends only on λ, and α(a, b) is the beta function. In particular, the exact value of a Chung-type law of the iterated logarithm established by Duker et al. (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 Jm,α(BH) are established to show the limit inferior behaviors. Similar Chung-type laws of the iterated logarithm and small ball probabilities for a Riemann-Liouville fractional process are also established.
