Large deviations of inverse processes with nonlinear scalings

Let Z be a two-dimensional Brownian motion confined to the non-negative quadrant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for two-station queueing networks. The parameters of Z are a drift vector, a covariance matrix, and a "direction of reflection" for each of the quadrant's two boundary rays. Necessary and sufficient conditions are known for Z to be a positive recurrent semimartingale, and they are the only restrictions imposed on the process data in our study. Under those assumptions, a large deviations principle (LDP) is conjectured for the stationary distribution of Z, and we recapitulate the cases for which it has been rigorously justified. For sufficiently regular sets B, the LDP says that the stationary probability of xB decays exponentially as x??, and the asymptotic decay rate is the minimum value achieved by a certain function I(?) over the set B. Avram, Dai and Hasenbein (Queueing Syst.: Theory Appl. 37, 259---289, 2001) provided a complete and explicit solution for the large deviations rate function I(?). In this paper we re-express their solution in a simplified form, showing along the way that the computation of I(?) reduces to a relatively simple problem of least-cost travel between a point and a line.

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.

This paper deals with a random walk $S_n:=\xi_1+\cdots+\xi_n$, $n=0,1,\ldots,$ in the $d$-dimensional Euclidean space ${\mathbb R}^d$, where $S_0=0$ and $\xi_k$ are independent identically distributed random vectors satisfying Cramér's moment conditions. For random polygons with nodes at the points $(\frac{k}{n},\frac{1}{x}S_k)$, $k=0,1,\ldots,n,$ we obtain the logarithmic asymptotics of the large deviation probabilities in different trajectory spaces when $x\sim \alpha_0 n$, $\alpha_0>0$, as $n\to\infty.$ The results include the so-called local and extended large deviation principles (l.d.p.'s) (see i̧te15) that hold in those cases where the “usual” l.d.p. does not apply. The paper consists of three parts. Part I has two sections. Section 1 presents the key concepts and some facts concerning the l.d.p. in arbitrary metric spaces. In section 2 we formulate the “strong” versions of the “usual” l.d.p. in the large deviation zones that were obtained earlier in [A. A. Borovkov, Theory Probab. Appl., 12 (1967), pp. 575--595], [A. A. Mogul'skii, Theory Probab. Appl., 21 (1976), pp. 300--315] for the space of continuous functions. Besides that, section 2 also contains the l.d.p. for probabilities for the random walk trajectories to hit a convex set. That result was obtained using inequalities from [A. A. Borovkov and A. A. Mogul'skii, Theory Probab. Appl., 56 (2012), pp. 21--43] and does not involve any moment conditions. Part II begins with section 3 presenting an example elucidating the need to extend both the problem formulation and the very concept of the “large deviation principle.” We introduce a new extended functional space, a metric therein, and the deviation functional (integral) of a more general kind that will be used when constructing an “extended” l.d.p. In section 4 we present and prove the key results of the paper, the local and the extended large deviation principles, for the trajectories of univariate random walks in the space ${\mathbb D}$ of functions without discontinuities of the second kind. Section 5 extends to the multivariate case all the results established in section 4. Section 6 in Part III presents results analogous to those from section 4, but now established in the space of functions of bounded variation with a metric stronger than that in $\mathbb D$. In section 7 we establish the so-called conditional large deviation principles for the trajectories of univariate random walks given the location of the walk at the terminal point. As a consequence, we obtain the Sanov's theorem on large deviations of empirical distributions.

We show, under regularity conditions, that a counting process satisfies a large deviations principle in ℝ or the Gartner-Ellis condition (convergence of the normalized logarithmic moment generating functions) if and only if its inverse process does. We show, again under regularity conditions, that embedded regenerative structure is sufficient for the counting process or its inverse process to have exponential asymptotics, and thus satisfy the Gartner-Ellis condition. These results help characterize the small-tail asymptotic behavior of steady-state distributions in queueing models, e.g., the waiting time, workload and queue length.

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.

This paper studies products of independent but non-identically distributed random network matrices that arise as weight matrices in distributed consensus-type computation and inference procedures in peer-to-peer multi-agent networks. The non-identically distributed matrices studied in this paper model various application scenarios in which the agent communication network is time-varying, either naturally or engineered to achieve communication efficiency in computational procedures. First, under broad conditions on the statistics of the network matrix sequence, the product of the sequence is shown to converge almost surely to the consensus matrix and explicit large deviations rate of convergence are obtained. Specifically, given the admissible graph of interconnections modeling the base network topology, it is shown that the large deviations rate of consensus equals the minimum limiting value of the fluctuating graph cuts, where the edge costs are assigned through the current probabilities of the inter-agent communications. Secondly, an application of the above large deviations principle is studied in the context of distributed detection in time-varying networks with sequential observations. By adopting a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">consensus+innovations</i> type distributed detection algorithm, as a by-product of this result, error exponents are obtained for the performance of distributed detection. It is shown that slow starts (slow increase) of inter-agent communication probabilities yield the same asymptotic error rate – and hence the same distributed detection performance, as if the communications were at their nominal levels from the beginning. As an important special case it is shown that when all the intermittent graph cuts have a link the probability of which increases to one, the performance of distributed detection is asymptotically optimal - i.e., equivalent to a centralized setup having access to all network data at all times.

Large deviations for some non-standard telegraph processes

Strong large deviations principles for a general class of discrete generation and interacting particle systems are developed. The analysis is essentially conducted through an original projective interpretation of the τ-topology, combined with a powerful projective transfer result presented by the first author with J. Gartner. These large deviations principles simplify and encompass the ones obtained in an earlier joint work of the second author with A. Guionnet. They are illustrated with simplified versions of McKean-Vlasov diffusions, and Boltzmann type collision models. We also describe the impact of this analysis on a recently developed class of genealogical and interacting particle interpretations of non linear Feynman-Kac-Schrodinger path measures.

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 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).

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.

We prove the large deviation principle for the trajectory of a broad class of mean field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates. The main step in the proof of the large deviation principle, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations. Additionally, we show that the large deviation principle provides a general method to identify a Lyapunov function for the associated McKean-Vlasov equation.

In this paper, we consider solutions of stochastic equations with small diffusion and a small jump function. The equation coefficients are random and depend nonregularly on a small parameter. The principle of large deviations is proved. The paper studies large deviations in the averaging scheme for various assumptions on the character of a disturbing process.