Large deviations for stochastic flows of diffeomorphisms

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.

This paper concerns the large deviations regime and the consequent solution of the Kramers problem for a two-time scale stochastic system driven by a common jump noise signal perturbed in small intensity [Formula: see text] and with accelerated jumps by intensity [Formula: see text]. We establish Freidlin–Wentzell estimates for the slow process of the multiscale system in the small noise limit [Formula: see text] using the weak convergence approach to large deviations theory. The core of our proof is the reduction of the large deviations principle to the establishment of a stochastic averaging principle for auxiliary controlled processes. As consequence we solve the first exit time/exit locus problem from a bounded domain containing the stable state of the averaged dynamics for the family of the slow processes in the small noise limit.

In this chapter we use the representations derived in Chap. 8 to study large and moderate deviations for stochastic systems driven by Brownian and/or Poisson noise, and consider a “small noise” limit, as in Sects. 3.2 and 3.3. We will prove general abstract large deviation principles, and in later chapters apply these to models in which the noise enters the system in an additive and independent manner (In our terminology, this includes systems with multiplicative noise, namely settings in which the noise term is multiplied by a state-dependent coefficient). For these systems, one can view the mapping that takes the noise into the state of the system as “nearly” continuous, and it is this property that allows a unified and relatively straightforward treatment.

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.

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

Large Deviations of Realized Volatility

The large deviation principle is established for the Poisson-Dirichlet distribution when the parameterapproaches infinity. The result is then used to study the asymptotic behavior of the homozygosity and the Poisson-Dirichlet distribution with selection. A phase transition occurs depending on the growth rate of the selection intensity. If the selection intensity grows sublinearly in �, then the large deviation rate function is the same as the neutral model; if the selection inten- sity grows at a linear or greater rate in �, then the large deviation rate function includes an additional term coming from selection. The application of these results to the heterozygote advantage model pro- vides an alternate proof of one of Gillespie's conjectures in (Theoret. Popul. Biol. 55 145-156).

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 the large deviations principles (LDPs) for the stochastic linear Schrödinger equation and its symplectic discretizations. These numerical discretizations are the spatial semi-discretization based on the spectral Galerkin method, and the further full discretizations with symplectic schemes in temporal direction. First, by means of the abstract Gärtner–Ellis theorem, we prove that the observable $B_{T}=\frac {u(T)}{T}$ , T > 0 of the exact solution u is exponentially tight and satisfies an LDP on $L^{2}(0, \pi ; \mathbb C)$ . Then, we present the LDPs for both $\{{B^{M}_{T}}\}_{T>0}$ of the spatial discretization $\{u^{M}\}_{M\in \mathbb N}$ and $\{{B^{M}_{N}}\}_{N\in \mathbb N}$ of the full discretization $\{{u^{M}_{N}}\}_{M,N\in \mathbb N}$ , where ${B^{M}_{T}}=\frac {u^{M}(T)}{T}$ and ${B^{M}_{N}}=\frac {{u^{M}_{N}}}{N\tau }$ are the discrete approximations of BT. Further, we show that both the semi-discretization $\{u^{M}\}_{M\in \mathbb N}$ and the full discretization $\{{u^{M}_{N}}\}_{M,N\in \mathbb N}$ based on temporal symplectic schemes can weakly asymptotically preserve the LDP of {BT}T> 0. These results show the ability of symplectic discretizations to preserve the LDP of the stochastic linear Schrödinger equation, and first provide an effective approach to approximating the large deviations rate function in infinite dimensional space based on the numerical discretizations.

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.

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).

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 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.

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.