Related papers: A uniform result for the dimension of fractional B…

We obtain bounds for probabilities of deviations of the truncated variation functional of fractional Brownian motions (fBm) of any Hurst index $H \in (0,1)$ from their expected values. Obtained bounds are optimal for large values of…

We study the long-time asymptotics of the probability P_t that the Riemann-Liouville fractional Brownian motion with Hurst index H does not escape from a fixed interval [-L,L] up to time t. We show that for any H \in ]0,1], for both…

Let $B^{H}$ be a $d$-dimensional fractional Brownian motion with Hurst index $H\in(0,1)$, $f:[0,1]\longrightarrow\mathbb{R}^{d}$ a Borel function, and $E\subset[0,1]$, $F\subset\mathbb{R}^{d}$ are given Borel sets. The focus of this paper…

We consider the integral of fractional Brownian motion (IFBM) and its functionals $\xi_T$ on the intervals $(0,T)$ and $(-T,T)$ of the following types: the maximum $M_T$, the position of the maximum, the occupation time above zero etc. We…

Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1/2$. In this paper we study the {\it generalized quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t=\lim_{\epsilon\downarrow…

We consider the paths of a Gaussian random process $x(t)$, $x(0)=0$ not exceeding a fixed positive level over a large time interval $(0,T)$, $T\gg 1$. The probability $p(T)$ of such event is frequently a regularly varying function at…

Let X^{1}, X^{2} be two independent (two-sided) fractional Brownian motions having the same Hurst parameter H in (0,1), and let Y be a standard (one-sided) Brownian motion independent of (X^{1},X^{2}). In dimension 2, fractional Brownian…

The frontier of a planar Brownian motion is the boundary of the unbounded component of the complement of its range. In this paper we find the Hausdorff dimension of the set of double points on the frontier.

Let $W^H=\{W^H(t), t \in \rr\}$ be a fractional Brownian motion of Hurst index $H \in (0, 1)$ with values in $\rr$, and let $L = \{L_t, t \ge 0\}$ be the local time process at zero of a strictly stable L\'evy process $X=\{X_t, t \ge 0\}$ of…

The present article is devoted to a fine study of the convergence of renormalized weighted quadratic and cubic variations of a fractional Brownian motion $B$ with Hurst index $H$. In the quadratic (resp. cubic) case, when $H<1/4$ (resp.…

Fractional Brownian motion, H-FBM , of index with d-dimensional time is considered in a spherical domain that contains 0 at its boundary. The main result : the log-asymptotics of probability that H-FBM does not exceed a fixed positive level…

The goal of this paper is to establish a relation between characteristic polynomials of $N\times N$ GUE random matrices $\mathcal{H}$ as $N\to\infty$, and Gaussian processes with logarithmic correlations. We introduce a regularized version…

Robert Kaufman's proof that the set of rapid points of Brownian motion has a Fourier dimension equal to its Hausdorff dimension was first published in 1974. A study of the proof of the original paper revealed several gaps in the arguments…

Let $B_H=\{B_H(t):t\in\mathbb R\}$ be a fractional Brownian motion with Hurst parameter $H\in(0,1)$. For the stationary storage process $Q_{B_H}(t)=\sup_{-\infty<s\le t}(B_H(t)-B_H(s)-(t-s))$, $t\ge0$, we provide a tractable criterion for…

Uniform large deviation principles for positive functionals of all equivalent types of infinite dimensional Brownian motions acting together with a Poisson random measure are established. The core of our approach is a variational…

In this paper, we show how concentration inequalities for Gaussian quadratic form can be used to propose exact confidence intervals of the Hurst index parametrizing a fractional Brownian motion. Both cases where the scaling parameter of the…

In this paper, firstly, we generalize the definition of the bifractional Brownian motion $B^{H,K}:=\Big(B^{H,K}\;;\;t\geq 0\Big)$, with parameters $H\in(0,1)$ and $K\in(0,1]$, to the case where $H$ is no longer a constant, but a function…

We show that the almost sure $\theta$-intermediate dimension of the image of the set $F_p =\{0, 1,\frac{1}{2^p},\frac{1}{3^p},\ldots\}$ under index-$h$ fractional Brownian motion is $\frac{\theta}{ph+\theta}$, a value that is smaller than…

We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate…

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$ for all…