Large Deviation Principles for Random Walk Trajectories. I

The present paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 56 (2012), pp. 538--561]. It consists of three sections. Section 3 presents an example showing that it is necessary to extend the problem setup and the very concept of the "large deviations principle" (l.d.p.). We introduce a new, extended function space, a metric in it, and a deviation functional (integral) of a more general form (compared to the usual one) that will be used to construct an "extended'' l.d.p. In section 4 we present and prove the main results of the paper for trajectories of univariate random walks in the space ${\mathbb D}$ of functions without discontinuities of the second kind: the local and extended l.d.p.'s. Section 5 extends all the results from section 4 to the multivariate case.

The scaling limits for d-dimensional random walks perturbed by an attractive force toward the origin are studied under the critical situation that the rate functional of the corresponding large deviation principle admits two minimizers. Our results extend those obtained by (2) from the mean-zero Gaussian to non-Gaussian setting under the absence of the wall. 1. Introduction and main result. It is a general principle in the study of various kinds of scaling limits that the limit points, at least at the level of law of large numbers, can be characterized by variational problems which minimize the rate functionals of the corresponding large deviation principles. However, if the rate functional admits several minimizers, the large deviation principle is not sucient to give an appropriate answer. This paper discusses such problem, especially for random walks on R d perturbed by an attractive force toward the origin 0 2 R d , motivated by certain models for interfaces or directed polymers.

Thus far we have a quite satisfactory theory of large deviations for Markov processes under the assumption (6.1), in case (D), or (6.8), in case (C). Unfortunately, when E is not compact, such an assumption will not be satisfied, except in very special situations (e.g. the Sanov theorem). For example, let E = R1 and P(t,x,dy) = g(1-e-2t,y-e-tx)dy, where g(s, ξ) = (2πs)-1/2 exp(-ξ2/2s). Clearly, the associated process (i.e. the Ornstein-Uhlenbeck process) has strong ergodic properties and one ought to be able to study the large deviation theory. At the same time, it is equally clear that P(t,x,•) fails to satisfy (6.8), and therefore the theory developed in section 6) is not applicable. Of course, the reason why section 6) cannot handle this process is clear; namely: when section 6) applies, the resulting large deviation principle is uniform, whereas one should not expect a uniform principle to hold for the Ornstein-Uhlenbeck process. Indeed, one can hope for uniform large deviation principles only in the presence of uniform ergodicity.

We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser. Fiz.--Mat. Nauk 6 (1969) 17--22, Theory Probab. Appl. 14 (1969) 51--64, 193--208] on large deviations for sums of i.i.d. regularly varying random variables to partial sum processes of i.i.d. regularly varying vectors. The results are stated in terms of a heavy-tailed large deviation principle on the space of c\`{a}dl\`{a}g functions. We illustrate how these results can be applied to functionals of the partial sum process, including ruin probabilities for multivariate random walks and long strange segments. These results make precise the idea of heavy-tailed large deviation heuristics: in an asymptotic sense, only the largest step contributes to the extremal behavior of a multivariate random walk.

We prove a {\it{quenched}} large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster (SRWPC) on $\mathbb Z^d$ ($d\geq 2$). The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and {the vacant set of random interlacements} (for $d\geq 3$) and the level sets of the Gaussian free field ($d\geq 3$). Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz ([Y08]) and Rosenbluth ([R06]) for similar results regarding elliptic random walks in random environment, we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the {\it{pair empirical measures}} of the environment Markov chain in the non-elliptic case of SRWPC . Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas. The main difficulty in our set up lies in the inherent non-ellipticity as well as the lack of {\it{translation-invariance}} stemming from conditioning on the fact that the origin belongs to the infinite cluster. We develop a unifying approach for proving quenched large deviations for SRWPC based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster.

We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments.We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments. However, line segments may not be optimal in general.Furthermore, we find explicitly the rate function of the area of the convex hull for random walks with rotationally invariant distribution of increments. For such walks, which necessarily have zero mean, large deviations of the area are attained by the trajectories that asymptotically align into half-circles. For random walks with non-zero mean increments, we find the rate function of the area for Gaussian walks with drift. Here the optimal limit shapes are elliptic arcs if the covariance matrix of increments is non-degenerate and parabolic arcs if otherwise.The above results on convex hulls of Gaussian random walks remain valid for convex hulls of planar Brownian motions of all possible parameters. Moreover, we extend the LDPs for the perimeter and the area of convex hulls to general Lévy processes with finite Laplace transform.

This paper studies large deviations properties of vectors of empirical means and measures generated as follows. Consider a sequence X1,X2,…,Xn of independent and identically distributed random variables partitioned into d-subgroups with sizes n1,…,nd. Further, consider a d-dimensional vector mn whose coordinates are made up of the empirical means of the subgroups. We prove the following. (1) The sequence of vector of empirical means mn satisfies large deviations principle with rate n and rate function I, when the sequence X1,X2,…,Xn is Rl valued, with l≥1. (2) Similar large deviations results hold for the corresponding sequence of vector of empirical measures Ln if Xi’s, i=1,2,…,n, take on finitely many values. (3) The rate functions for the above large deviations principles are convex combinations of the corresponding rate functions arising from the large deviations principles of the coordinates of mn and Ln. The probability distributions used in the convex combinations are given by α=(α1,…,αd)=limn→∞1/n(n1,…,nd). These results are consequently used to derive variational formula for the thermodynamic limit for the pressure of multipopulation Curie-Weiss (I. Gallo and P. Contucci (2008), and I. Gallo (2009)) and mean-field Pott’s models, via a version of Varadhan’s integral lemma for an equicontinuous family of functions. These multipopulation models serve as a paradigm for decision-making context where social interaction and other socioeconomic attributes of individuals play a crucial role.

A large deviation principle is established for a general class of stochastic flows in the small noise limit. This result is then applied to a Bayesian formulation of an image matching problem, and an approximate maximum likelihood property is shown for the solution of an optimization problem involving the large deviations rate function.

Large deviation principles are established for some random\ncombinatorial structures including the Ewens sampling formula and the Pitman\nsampling formula. A path-level large deviation principle is established for the\nformer on the cadlag space D$(o, 1], R)$ equipped with the uniform convergence\ntopology, and the rate function is the same as for a Poisson process justifying\nthe Poisson process approximation for the Ewens sampling formula at the large\ndeviation level. A large deviation principle for the total number of parts in a\npartition is obtained for the Pitman formula; here the rate function depends\nonly on one of the two parameters which display the different roles of the two\nparameters at different scales. In addition to these large deviation results,\nwe also provide an embedding scheme which gives the Pitman sampling formula. A\nproduct of this embedding is an intuitive alternate proof of a result of Pitman\non the limiting total number of parts.

Moderate and large deviation principles for the hazard rate function kernel estimator under censoring

In this paper we prove a Sanov result, i.e. a Large Deviation Principle (LDP) for the distribution of the empirical measure, for the annealed Glauber dynamics of the Sherrington-Kirkpatrick spin-glass. Without restrictions on time or temperature we prove a full LDP for the asymmetric dynamics and the crucial upper large deviations bound for the symmetric dynamics. In the symmetric model a new order-parameter arises corresponding to the response function in [SoZi83]. In the asymmetric case we show that the corresponding rate function has a unique minimum, given as the solution of a self-consistent equation. The key argument used in the proofs is a general result for mixing of what is known as Large Deviation Systems (LDS) with measures obeying an independent LDP.

In this work, a Wentzell-Freidlin type large deviation principle is established for the two-dimensional stochastic Navier-Stokes equations (SNSE's) with nonlinear viscosities. We fi_x000C_rst prove the existence and uniqueness of solutions to the two-dimensionalstochastic Navier-Stokes equations with nonlinear viscosities using the martingale problem argument and the method of monotonicity. By the results of Varadhan and Bryc, the large deviation principle (LDP) is equivalent to the Laplace-Varadhan principle (LVP) if the underlying space is Polish. Then using the stochastic control and weak convergence approach developed by Budhiraja and Dupuis, the Laplace-Varadhan principle for solutions of stochastic Navier-Stokesequations is obtained in appropriate function spaces.

Previous article Next article Superlarge Deviation Probabilities for Sums of Independent Random Variables with Exponential Decreasing DistributionL. V. RozovskyL. V. Rozovskyhttps://doi.org/10.1137/S0040585X9798289XPDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIn this paper large deviation probabilities of sums of independent identically distributed random variables are studied, whose distribution function has an exponential decreasing tail.[1] L. V. Rozovsky, Superlarge deviations of a sum of independent random variables having a common absolutely continuous distribution under the Cramér condition, Theory Probab. Appl., 48 (2005), pp. 108–130. TPRBAU 0040-585X LinkGoogle Scholar[2] A. A. Borovkov and and B. A. Rogozin, On the multidimensional central limit theorem, Theory Probab. Appl., 10 (1965), pp. 55–62. TPRBAU 0040-585X LinkGoogle Scholar[3] L. V. Rozovsky, Large-deviation probabilities for some classes of distributions satisfying the Cramér condition, Zap. Nauch. Semin. POMI, 298 (2003), pp. 161–185 (in Russian). ZNSPEV 0136-1244 Google Scholar[4] L. V. Rozovsky, On a lower bound of large-deviation probabilities for the sample mean under the Cramér condition, Zap. Nauch. Semin. POMI, 278 (2001), pp. 208–224 (in Russian); J. Math. Sci. New York, 118 (2003), pp. 5624–5634 (in English). Google Scholar[5] A. A. Borovkov and and A. A. Mogulskii, On large and superlarge deviations of sums of independent random vectors under Cramér's condition. II, Theory Probab. Appl., 51 (2007), pp. 567–594. TPRBAU 0040-585X LinkGoogle ScholarKeywordsindependent random variableslarge deviationsregular varying function Previous article Next article FiguresRelatedReferencesCited byDetails Superlarge Deviation Probabilities for Sums of Independent Random Variables with Exponentially Decreasing Tails. II25 March 2015 | Theory of Probability & Its Applications, Vol. 59, No. 1AbstractPDF (171 KB) Volume 52, Issue 1| 2008Theory of Probability & Its Applications History Submitted:23 November 2005Published online:10 March 2008 InformationCopyright © 2008 Society for Industrial and Applied MathematicsKeywordsindependent random variableslarge deviationsregular varying functionPDF Download Article & Publication DataArticle DOI:10.1137/S0040585X9798289XArticle page range:pp. 167-171ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

This article studies large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an α-stable law, α∈(0,2], with emphasis on the case α=2. There are two different scenarios: either the deviation is realised via a collective behaviour with all summands contributing to the deviation (a Gaussian scenario), or a single summand is atypically large and contributes to the deviation (a one-big-jump scenario). Such results are known when α∈(0,2) (large deviations always follow a one big-jump scenario) or when the random variables admit a moment of order 2+δ for some δ>0. We extend these results, including in particular the case where the right tail is regularly varying with index -2 (treating cases with infinite variance in the domain of attraction of the normal law). We identify the threshold for the transition between the Gaussian and the one-big-jump regimes; it is slightly larger when considering local large deviations compared to integral large deviations. Additionally, we complement our results by describing the behaviour of the sum and of the largest summand conditionally on a (local) large deviation, for any α∈(0,2], both in the Gaussian and in the one-big-jump regimes. As an application, we show how our results can be used in the study of condensation phenomenon in the zero-range process at the critical density, extending the range of parameters previously considered in the literature.

In this paper, we consider McKean-Vlasov stochastic differential equations (MVSDEs) driven by Lévy noise. By identifying the right equations satisfied by the solutions of the MVSDEs with shifted driving Lévy noise, we build up a framework to fully apply the weak convergence method to establish large and moderate deviation principles for MVSDEs. In the case of ordinary SDEs, the rate function is calculated by using the solutions of the corresponding skeleton equations simply replacing the noise by the elements of the Cameron-Martin space. It turns out that the correct rate function for MVSDEs is defined through the solutions of skeleton equations replacing the noise by smooth functions and replacing the distributions involved in the equation by the distribution of the solution of the corresponding deterministic equation (without the noise). This is somehow surprising. With this approach, we obtain large and moderate deviation principles for much wider classes of MVSDEs in comparison with the existing literature see Dos Reis et al. (Ann. Appl. Probab. 29, 1487–1540, 2019).