Large deviations of degenerate Mises functionals

Ya. Yu. Nikitin's method of obtaining asymptotics of large deviations is extended to a more general situation. Bibliography:14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 256–261.     

Large deviations for renewal processes

We investigate large deviations for the empirical measure of the forward and backward recurrence time processes associated with a classical renewal process with arbitrary waiting-time distribution. The Donsker-Varadhan theory cannot be applied in this case, and indeed it turns out that the large deviations rate functional differs from the one suggested by such a theory. In particular, a non-strictly convex and non-analytic rate functional is obtained.     

Large deviations for functionals of spatial point processes with applications to random packing and spatial graphs

Functionals of spatial point process often satisfy a weak spatial dependence condition known as stabilization. We prove general Donsker–Varadhan large deviation principles (LDP) for such functionals and show that the general result can be applied to prove LDPs for various particular functionals, including those concerned with random packing, nearest neighbor graphs, and lattice versions of the Voronoi and sphere of influence graphs.     

Large deviations for almost markovian processes

Summary Large deviation theorems for the empirical distribution of almost markovian processes are proven. Relative entropy identifies the rate function and its definition depends only on the process.Work partially supported by grant NSF-DMR81-14726     

Large deviations for conditional Volterra processes

In this article we investigate a problem of large deviations for continuous Gaussian Volterra processes, conditioned to follow a fixed trajectory up to a fixed time T > 0, in order to establish the behavior of the process in the near future after T and to give an asymptotic estimate of the exit probability of its bridge. Some examples are considered.     

LARGE DEVIATIONS FOR FIELDS WITH STATIONARY INDEPENDENT INCREMENTS

ＬＡＲＧＥＤＥＶＩＡＴＩＯＮＳＦＯＲＦＩＥＬＤＳＷＩＴＨＳＴＡＴＩＯＮＡＲＹＩＮＤＥＰＥＮＤＥＮＴＩＮＣＲＥＭＥＮＴＳＧＡＯＦＵＱＩＮＧ（高付清）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕｂｅｉＵｎｉｖｅｒｓｉｔｙ，Ｗｕｈａｎ４３００７２...     

Large deviation for stationary processes

A quasi-local variational characterization of the entropy for stationary processes is given. This is used to establish upper and lower large deviation estimates for arbitrary stationary processes. The upper and lower rate functions are shown to coincide for all quasi-local stationary processes. The contents of the paper is the following: 1. Introduction; 2. Notations; 3. Relative entropy of conditional expectations; 4. Relative entropy of a stationary process with respect to a covariant family of conditional expectations; 5. The role of locality and quasi-locality properties; 6. Large deviation upper estimate; 7. The Lower estimate; 8. The variational principle.     

Large deviations of renewal processes

Limit theorems for large deviations of renewal processes are presented. One result is for a terminating renewal process with small probability of terminating. These theorems are analogous to the classical Cramer and Feller large deviation theorems for sums of independent random variables.     

Large deviations for perturbed reflected diffusion processes

In this article, we establish a large deviation principle for the solutions of perturbed reflected diffusion processes. The key is to prove a uniform Freidlin–Ventzell estimate of perturbed diffusion processes.     

Moderate deviations for quadratic forms in Gaussian stationary processes

Moderate deviations limit theorem is proved for quadratic forms in zero-mean Gaussian stationary processes. Two particular cases are the cumulative periodogram and the kernel spectral density estimator. We also derive the exponential decay of moderate deviation probabilities of goodness-of-fit tests for the spectral density and then discuss intermediate asymptotic efficiencies of tests.     

Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes

Let X(t), , be a centered real-valued stationary Gaussian process with spectral density f(λ). The paper considers a question concerning asymptotic distribution of Toeplitz type quadratic functional Q _T of the process X(t), generated by an integrable even function g(λ). Sufficient conditions in terms of f(λ) and g(λ) ensuring central limit theorems for standard normalized quadratic functionals Q _T are obtained, extending the results of Fox and Taqqu (Prob. Theory Relat. Fields 74: 213–240, 1987), Avram (Prob. Theory Relat. Fields 79:37–45, 1988), Giraitis and Surgailis (Prob. Theory Relat. Fields 86: 87–104, 1990), Ginovian and Sahakian (Theory Prob. Appl. 49:612–628, 2004) for discrete time processes.      

Central limit theorems for nonlinear functionals of stationary Gaussian processes

Let X=(X _t ,t) be a stationary Gaussian process on (, ,P), letH(X) be the Hilbert space of variables inL ² (,P) which are measurable with respect toX, and let (U _s ,s) be the associated family of time-shift operators. We sayYH(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if in [respectively,], where

Estimation error for occupation time functionals of stationary Markov processes

The approximation of integral functionals with respect to a stationary Markov process by a Riemann sum estimator is studied. Stationarity and the functional calculus of the infinitesimal generator of the process are used to explicitly calculate the estimation error and to prove a general finite sample error bound. The presented approach admits general integrands and gives a unifying explanation for different rates obtained in the literature. Several examples demonstrate how the general bound can be related to well-known function spaces.