Kavli Affiliate: Ran Wang
| First 5 Authors: Ran Wang, Yimin Xiao, , ,
| Summary:
Let $X:={X(t)}_{tge0}$ be a generalized fractional Brownian motion: $$
{X(t)}_{tge0}overset{d}{=}left{ int_{mathbb R}
left((t-u)_+^{alpha}-(-u)_+^{alpha} right) |u|^{-gamma/2} B(du)
right}_{tge0}, $$ with parameters $gamma in (0, 1)$ and $alphain
left(-1/2+ gamma/2, , 1/2+ gamma/2 right)$. This is a self-similar
Gaussian process introduced by Pang and Taqqu (2019) as the scaling limit of
power-law shot noise processes. The parameters $alpha$ and $gamma$ determine
the probabilistic and statistical properties of $X$. In particular, the
parameter $gamma$ introduces non-stationarity of the increments. In this
paper, we prove Strassen’s local law of the iterated logarithm of $X$ at any
fixed point $t_0 in (0, infty)$, which describes explicitly the roles played
by the parameters $alpha, gamma$ and the location $t_0$. Our result is
different from the previous Strassen’s LIL for $X$ at infinity proved by
Ichiba, Pang and Taqqu (2022).
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