Small Deviation Asymptotics Research Articles

On Small Deviation Asymptotics in the L2-Norm for Certain Gaussian Processes

The results obtained allow finding sharp small deviations in a Hilbert norm for centered Gaussian processes in the case where their covariances have a special form of the eigenvalues and allow us to describe small deviation asymptotics for certain Gaussian processes.

On the Exact Asymptotics of Small Deviations of $L_2$-Norm for some Gaussian Random Fields

In this paper we study the asymptotic behavior of the tail probability $P(V^2<r)$ as $r\to 0$, where the sum $V^2$ is given by the formula $ V^2=a^2 \sum_{i,j\ge 1} (i+\beta)^{-2c}(j+\delta)^{-2}\xi^2_{ij}. $ Here $\{\xi_{ij}\}$ are independent standard Gaussian random variables, and $a>0$, $\beta >-1$, $\delta>-1$, $c>1/2$, $\ne 1$ are some constants. Thus, we study small deviations of the $L_2$-norm of certain two-parameter Gaussian random fields, that have the structure of a tensor product.

On Small Deviation Asymptotics In L2 of Some Mixed Gaussian Processes

We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case and in some examples of more complicated processes of similar structure. The proof is based on Karhunen–Loève expansion together with spectral asymptotics of differential operators and complex analysis methods.

Precise small deviations in L 2 of some Gaussian processes appearing in the regression context

Abstract We find precise small deviation asymptotics with respect to the Hilbert norm for some special Gaussian processes connected to two regression schemes studied by MacNeill and his coauthors. In addition, we also obtain precise small deviation asymptotics for the detrended Brownian motion and detrended Slepian process.

Limit theorems for small deviation probabilities of some iterated Stochastic processes

We derive logarithmic asymptotics for probabilities of small deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those of the processes which generate the iterated processes. When these conditions do not hold, the asymptotics of small deviations for iterated processes are quite different. We apply our results to the iterated processes generated by the compound Cox processes and compound renewal processes. Bibliography: 10 titles.

Ergodic means for large values of and exact asymptotics of small deviations for a multi-dimensional Wiener process

We prove results on exact asymptotics as for the means and probabilities , where , , is an -dimensional Wiener process, is a positive continuous function (potential) satisfying certain conditions, , and are prescribed vectors. The results are obtained by a new method developed in this paper, the Laplace method for the occupation time of a multi-dimensional Wiener process. We consider examples of monomial and radial potentials and prove results on exact asymptotics of small deviations for the probabilities as with a fixed .

Exact Small Deviation Asymptotics for Some Brownian Functionals

We find exact small deviation asymptotics with respect to a weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require knowledge of the eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.

Small Deviations Asymptotics for Matérn Processes and Fields under Weighted Quadratic Norm

We prove results on an exact small deviation asymptotics in $L_2$-norm for Matern processes with arbitrary natural index and on logarithmic asymptotics for Matern processes and fields with arbitrary indexes.

Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm

We obtain results on small deviations of Bogoliubov’s Gaussian measure occurring in the theory of the statistical equilibrium of quantum systems. For some random processes related to Bogoliubov processes, we find the exact asymptotic probability of their small deviations with respect to a Hilbert norm.

Exact asymptotics of small deviations for a stationary Ornstein-Uhlenbeck process and some Gaussian diffusion processes in the L p -norm, 2 ≤ p ≤ ∞

We prove results on exact asymptotics of the probabilities $$ P\left\{ {\int\limits_0^1 {\left| {\eta (t)} \right|^p dt \leqslant \varepsilon ^p } } \right\},\varepsilon \to 0, $$ where 2 ? p ? ?, for two types of Gaussian processes ?(t), namely, a stationary Ornstein-Uhlenbeck process and a Gaussian diffusion process satisfying the stochastic differential equation $$ \left\{ \begin{gathered} dZ(t) = dw(t) + g(t)Z(t)dt,t \in [0,1], \hfill \\ Z(0) = 0. \hfill \\ \end{gathered} \right. $$ Derivation of the results is based on the principle of comparison with a Wiener process and Girsanov's absolute continuity theorem.

Occupation times and exact asymptotics of small deviations of Bessel processes for -norms with

We prove theorems on exact asymptotics of the distributions of integral functionals of the occupation time of Bessel processes. Using these results, we obtain exact asymptotics of small deviations for Bessel processes in the -norm. We use Laplace's method for the occupation times of Markov processes with continuous time. Computations are carried out for and . We also solve extremal problems for the action functional.

On probabilities of small deviations for compound cox processes

We derive logarithmic asymtotics for probabilities of small deviations for compound Cox processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables and processes with independent increments. When these conditions do not hold, the asymptotics of small deviations for compound Cox processes are quite different. Bibliography: 9 titles.

Exact small deviation asymptotics for the Slepian and Watson processes in the Hilbert norm

We find the exact asymptotic behavior of small ball probabilities in the Hilbert norm for the simplest form of the Slepian process and for the Watson process appearing in nonparametric statistics. Bibliography: 23 titles.

Sharp Small Deviation Asymptotics in the L 2-Norm for a Class of Gaussian Processes

We characterize the exact behavior of small deviations in a Hilbert norm for centered Gaussian processes in the case where their covariances have a special form of eigenvalues. This result enables us to describe small deviation asymptotics for certain Gaussian processes. Bibliography: 20 titles.

Exact Small Ball Constants for Some Gaussian Processes under the L 2-Norm

We find some logarithmic and exact small deviation asymptotics for the L 2-norms of certain Gaussian processes closely connected with a Wiener process. In particular, processes obtained by centering and integrating Brownian motion and Brownian bridge are examined. Bibliography: 28 titles.

Exact L 2 -small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

We find the exact small deviation asymptotics for the L2-norm of various m-times integrated Gaussian processes closely connected with the Wiener process and the Ornstein – Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein – Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.