On large deviations for some sequences of weighted means of Gaussian processes

In this paper we study some sequences of weighted means of continuous real valued Gaussian processes. More precisely we consider suitable generalizations of both arithmetic and logarithmic means of a Gaussian process with covariance function which satisfies either an exponential decay condition or a power decay condition. Our aim is to provide limits of variances of functionals of such weighted means which allow the application of some large deviation results in the literature.     

On asymptotic behavior for probabilities of large deviations of compound Cox processes

We derive logarithmic asymptotics for probabilities of large 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 fail, the asymptotics of large deviations probabilities for compound Cox processes are quite different. Bibliography: 5 titles. Translated from Zapiski Nauehnykh, Seminarov POMI, Vol. 361, 2008, pp. 167–181.     

On large deviations in testing simple hypotheses for locally stationary Gaussian processes

We derive a large deviation result for the log-likelihood ratio for testing simple hypotheses in locally stationary Gaussian processes. This result allows us to find explicitly the rates of exponential decay of the error probabilities of type I and type II for Neyman?CPearson tests. Furthermore, we obtain the analogue of classical results on asymptotic efficiency of tests such as Stein??s lemma and the Chernoff bound, as well as the more general Hoeffding bound concerning best possible joint exponential rates for the two error probabilities.     

On the small time large deviations of diffusion processes on configuration spaces

In this paper, we establish a small time large deviation principle for diffusion processes on configuration spaces.     

On large deviations for approximations of SDEs

 We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of an SDE is close to that for an exact solution.     

On large deviations for the parabolic Anderson model

The focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic Anderson model above the mean versus large deviations below the mean. The functionals we treat are the solution u(x, t) to the spatially discrete parabolic Anderson model and a functional A _n which is used in analyzing the a.s. Lyapunov exponent for u(x, t). Both satisfy a “law of large numbers”, with ${\lim_{t\to \infty} \frac{1}{t} \log u(x,t)=\lambda (\kappa)}$ and ${\lim_{n\to \infty} \frac{A_n}{n}=\alpha}$ . We then think of αn and λ(κ)t as being the mean of the respective quantities A _n and log u(t, x). Typically, the large deviations for such functionals exhibits a strong asymmetry; large deviations above the mean take on a different order of magnitude from large deviations below the mean. We develop robust techniques to quantify and explain the differences.     

On probabilities of large deviations for UH-statistics

Translated from Lietuvos Matematikos Rinkinys, Vol. 35, No. 2, pp. 141–151, April–June, 1995.     

On asymptotic behavior of probabilities of large and moderate deviations for some iterated stochastic processes

We derive logarithmic asymptotics for probabilities of large deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables. When these conditions do not hold, the asymptotics of large deviations for iterated processes are quite different. When the iterated process is a homogeneous process with independent increments in which time is replaced by random one, the behavior of large and moderate deviations is studied in the case of finite variance. For this case, the following one-sided moment restriction are considered: the Cramèr condition, the Linnik condition, and the existence of moment of order p > 2 for a positive part. Bibliography: 6 titles.     

Lower large deviations for supercritical branching processes in random environment

Branching processes in random environment (Z _n : n ≥ 0) are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of Z, which means the asymptotic behavior of the probability {1 ≤ Z _n ≤ exp(nθ)} as n → ∞. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where ?(Z ₁ = 0 | Z ₀ = 1) > 0 and also much weaker moment assumptions.     

On the existence of smooth densities for jump processes

Summary We consider a Lévy processX _t and the solutionY _t of a stochastic differential equation driven byX _t; we suppose thatX _t has infinitely many small jumps, but its Lévy measure may be very singular (for instance it may have a countable support). We obtain sufficient conditions ensuring the existence of a smooth density forY _t: these conditions are similar to those of the classical Malliavin calculus for continuous diffusions. More generally, we study the smoothness of the law of variablesF defined on a Poisson probability space; the basic tool is a duality formula from which we estimate the characteristic function ofF.     

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. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 163–175.     

On large deviations and density functions

Suppose thatX ₁,X ₂, ... is a sequence of absolutely continuous or integer valued random variables with corresponding probability density functionsf _n (x). Let {φ _n } _n=1 ^∞ be a sequence of real numbers, then necessary and sufficient conditions are given forn ⁻¹ logf _n (φ _n )-n ⁻¹ log P (X _n >φ _n )=0(1) asn→∞.     

Inequalities and principles of large deviations for the trajectories of processes with independent increments

We consider a homogeneous process S(t) on [0,∞) with independent increments, establish the local and ordinary large deviation principles for the trajectories of the processes $s_T (t): = \tfrac{1} {T}S(tT) $ , t ∈ [0, 1], as T → ∞, and obtain a series of inequalities for the distributions of the trajectories of S(t).     

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(X ^ε (t)): ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψ_ε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X ε (t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψ_ε(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of Ψ_ε(u). AMS Subject Classifications 60F10, 91B30 This work has been partially supported by Murst Project “Metodi Stocastici in Finanza Matematica”     

Criteria for large deviations

We give the general variational form of

for any bounded above Borel measurable function on a topological space , where is a net of Borel probability measures on , and a net in converging to . When is normal, we obtain a criterion in order to have a limit in the above expression for all continuous bounded, and deduce new criteria of a large deviation principle with not necessarily tight rate function; this allows us to remove the tightness hypothesis in various classical theorems.

     

Notes on large deviations for branching processes indexed by a Poisson process

Lithuanian Mathematical Journal - Consider a continuous-time process {ZNt}, where {Zn} is a Galton–Watson process with offspring mean m, and {Nt} is a Poisson process independent of {Zn}. It...     

On large deviations of Kolmogorov-Smirnov-Renyi type statistics

One result of Smirnov's important paper [Uspehi Mat. Nauk.10, 179–206, (in Russian)] yields exponential bounds for the large deviations of his one-sided Smirnov statistic and the two-sided Kolmogorov statistic. In the present paper exponential bounds are given for the large deviations of a wide class of Kolmogorov-Smirnov-Renyi type statistics. As a by-product, exponential bounds for the large deviations of the corresponding limit distributions are obtained.