Small values of Gaussian processes and functional laws of the iterated logarithm (original) (raw)

Logarithmic L2-Small Ball Asymptotics for some Fractional Gaussian Processes

Theory of Probability & Its Applications, 2005

We find the logarithmic L 2 -small ball asymptotics of some Gaussian processes related to the fractional Brownian motion, fractional Ornstein -Uhlenbeck process (fOU) and their integrated analogues. To that end we use general theorems on spectral asymptotics of integral operators obtained by combining them with the classical theorem of Weyl. In the simplest case of fractional Brownian motion we generalize the result of . We consider also the fractional Lévy's Brownian motion as well as the multiparameter fOU process on the bounded domain.

Path Properties of a Generalized Fractional Brownian Motion

Journal of Theoretical Probability, 2021

The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and non-stationary noises with a power-law variance function. In this paper we study sample path properties of the generalized fractional Brownian motion, including Hölder continuity, path differentiability/non-differentiability, and functional and local Law of the Iterated Logarithms.

On the Generalized Fractional Brownian Motion

Mathematical Models and Computer Simulations, 2018

We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. We study its main stochastic properties and some increments characteristics. As an application, we deduce the properties of nonsemimartingality, Hölder continuity, nondifferentiablity, and existence of a local time.

LAN property for some fractional type Brownian motion

Arxiv preprint arXiv: …, 2011

Abstract: We study asymptotic expansion of the likelihood of a certain class of Gaussian processes characterized by their spectral density $ f_\ theta .Weconsiderthecasewhere. We consider the case where .Weconsiderthecasewhere f_\ theta\ PAR {x}\ sim_ {x\ to 0}\ ABS {x}^{-\ al (\ theta)} L_\ theta (x) $ with $ L_\ theta $ a ...

Some asymptotic results related to the law of iterated logarithm for Brownian motion

Journal of Theoretical Probability, 1995

For 0 < y < 1, let Ur(t) = t -I I~ 1 {B,>~I ds. The questions addressed in this paper are motivated by a result due to Strassen: almost surely, llm sup,_o~ Ur(t)= 1-exp{-4(7, -1-1)}. We show that Strassen's result is closely related to a large deviations principle for the family of random variables Ur(t), t>0. Also, when y=l, Ur(t)~O almost surely and we obtain some bounds on the rate of convergence. Finally, we prove an analogous limit theorem for discounted averages of the form 2Ig ~ D().s)lts,>~} ds as 2 $ 0, where D is a suitable discount function. These results also hold for symmetric random walks.

On Gaussian Processes Equivalent in Law to Fractional Brownian Motion

Journal of Theoretical Probability, 2004

We consider Gaussian processes that are equivalent in law to the fractional Brownian motion and their canonical representations. We prove a Hitsuda type representation theorem for the fractional Brownian motion with Hurst index H [ 1 2. For the case H > 1 2 we show that such a representation cannot hold. We also consider briefly the connection between Hitsuda and Girsanov representations. Using the Hitsuda representation we consider a certain special kind of Gaussian stochastic equation with fractional Brownian motion as noise.

Estimates for exponential functionals of continuous Gaussian processes with emphasis on fractional Brownian motion

arXiv (Cornell University), 2023

Our aim in this article is to provide explicit computable estimates for the cumulative distribution function (c.d.f.) and the p-th order moment of the exponential functional of a fractional Brownian motion (fBM) with drift. Using elementary techniques, we prove general upper bounds for the c.d.f. of exponential functionals of continuous Gaussian processes. On the other hand, by applying classical results for extremes of Gaussian processes, we derive general lower bounds. We also find estimates for the p-th order moment and the moment-generating function of such functionals. As a consequence, we obtain explicit lower and upper bounds for the c.d.f. and the p-th order moment of the exponential functionals of a fBM, and of a series of independent fBMs. In addition, we show the continuity in law of the exponential functional of a fBM with respect to the Hurst parameter.

Large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes

The Annals of Probability, 2011

In this paper, we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. We also show that a fractional Brownian motion and the related Riemann-Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional Brownian motion; (2) logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of Riemann-Liouville processes; (3) comparison of local and intersection local times based on embedding of a part of a fractional Brownian motion into the reproducing kernel Hilbert space of the Riemann-Liouville process.

Unilateral small deviations for the integral of fractional Brownian motion

Arxiv preprint math/0310413, 2003

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 ≫ 1. The probability p(T ) of such event is frequently a regularly varying function at ∞ with exponent θ. In applications this parameter can provide information on fractal properties of processes that are subordinate to x(·). For this reason the estimation of θ is an important theoretical problem. Here, we consider the process x(t) whose derivative is fractional Brownian motion with self-similarity parameter 0 < H < 1. For this case we produce new computational evidence in favor of the relations log p(T ) = −θ log T (1 + o(1)) and θ = H(1 − H). The estimates of θ are to within 0.01 in the range 0.1 ≤ H ≤ 0.9. An analytical result for the problem in hand is known for the markovian case alone, i.e., for H = 1/2. We point out other statistics of x(t) whose small values have probabilities of the same order as p(T ) in the log scale.