Existence of large deviations rate function for any S -unimodal map

Boundary-Value Problems for Random Walks and Large Deviations in Function Spaces

For an arbitrary non-renormalizable unimodal map of the interval,f:I→I, with negative Schwarzian derivative, we construct a related mapFdefined on a countable union of intervals Δ. For each interval Δ,Frestricted to Δ is a diffeomorphism which coincides with some iterate offand whose range is a fixed subinterval ofI. IfFsatisfies conditions of the Folklore Theorem, we callfexpansion inducing. Letcbe a critical point off. Forfsatisfyingf″(c) ≠ 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds forf: the ω-limit set of Lebesgue almost every point is the interval [f2,f(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.

We study a topologically exact, negative Schwarzian unimodal map without neutral periodic points whose critical point is non-recurrent and flat. Assuming that the critical order is polynomial or logarithmic, we establish the large deviation principle and provide a partial description of the minimizers of the rate function. We apply our main results to a certain parametrized family of unimodal maps in the same topological conjugacy class, and determine the sets of minimizers.

Chaotic communications schemes based on synchronization aim to provide security over the conventional communication schemes. Symbolic dynamics based on synchronization methods has provided high quality synchronization (5). Symbolic dynamics is a rigorous way to investigate chaotic behavior with finite precision and can be used combined with information theory (13). In previous works we have studied the kneading theory analysis of the Duffing equation (3) and the symbolic dynamics and chaotic synchronization in coupled Duffing oscillators (2) and (4). In this work we consider the complete synchronization of two identical coupled unimodal and bimodal maps. We relate the synchronization with the symbolic dynamics, namely, defining a distance between the kneading sequences generated by the map iterates in its critical points and defining n-symbolic synchronization. We establish the synchronization in terms of the topological entropy of two unidirectional or bidirectional coupled piecewise linear unimodal and bimodal maps. We also give numerical simulations with coupled Duffing oscillators that exhibit numerical evidence of the n-symbolic synchronization. Keywords: Chaotic synchronization, Symbolic dynamics, Symbolic synchronization, Kneading theory.

We study a sequence of symmetric $n$-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the $n$-player game is known to converge, as $n$ goes to infinity, to the unique equilibrium of an associated mean field game. Under suitable regularity conditions, in the absence of common noise, we complement this law of large numbers result with nonasymptotic concentration bounds for the Wasserstein distance between the $n$-player Nash equilibrium empirical measure and the mean field equilibrium. We also show that the sequence of Nash equilibrium empirical measures satisfies a weak large deviation principle, which can be strengthened to a full large deviation principle only in the absence of common noise. For both sets of results, we first use the master equation, an infinite-dimensional partial differential equation that characterizes the value function of the mean field game, to construct an associated McKean–Vlasov interacting $n$-particle system that is exponentially close to the Nash equilibrium dynamics of the $n$-player game for large $n$, by refining estimates obtained in our companion paper. Then we establish a weak large deviation principle for McKean–Vlasov systems in the presence of common noise. In the absence of common noise, we upgrade this to a full large deviation principle and obtain new concentration estimates for McKean–Vlasov systems. Finally, in two specific examples that do not satisfy the assumptions of our main theorems, we show how to adapt our methodology to establish large deviations and concentration results.

Planar embeddings of inverse limit spaces of unimodal maps

Under the scenario of high-frequency data, a consistent estimator of the realized Laplace transform of volatility is proposed by Todorov and Tauchen (Econometrica 80:1105–1127, 2012) and a related central limit theorem has been well established. In this paper, we investigate the asymptotic tail behaviour of the empirical realized Laplace transform of volatility (ERLTV). We establish both a large deviation principle and a moderate deviation principle for the ERLTV. The good rate function for the large deviation principle is well defined in the whole real space, which indicates a limit for the normalized logarithmic tail probability of the ERLTV. Moreover, we also derive the function-level large and moderate deviation principles for ERLTV.

In this article we find exponential good approximation of the empirical neigbourhood distribution of symbolled random graphs conditioned to a given empirical symbol distribution and empirical pair distribution. Using this approximation we shorten or simplify the proof of (Doku-Amponsah \& Morters, 2010, Theorem~2.5); the large deviation principle (LDP) for empirical neigbourhood distribution of symbolled random graphs. We also show that the LDP for the empirical degree measure of the classical Erd\H{o}s-R\'{e}nyi graph is a special case of (Doku-Amponsah \& Moerters, 2010, Theorem~2.5). From the LDP for the empirical degree measure, we derive an LDP for the the proportion of isolated vertices in the classical Erd\H{o}s-R\'{e}nyi graph.

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.

For a finite random graph, we defined a simple model of statistical\nmechanics. We obtain an annealed asymptotic result for the random partition\nfunction for this model on finite random graphs as n; the size of the graph is\nvery large. To obtain this result, we define the empirical bond distribution,\nwhich enumerates the number of bonds between a given couple of spins, and\nempirical spin distribution, which enumerates the number of sites having a\ngiven spin on the spinned random graphs. For these empirical distributions we\nextend the large deviation principle(LDP) to cover random graphs with\ncontinuous colour laws. Applying Varandhan Lemma and this LDP to the\nHamiltonian of the Ising model defined on Erdos-Renyi graphs, expressed as a\nfunction of the empirical distributions, we obtain our annealed asymptotic\nresult.\n

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical nonsteepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the estimator previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical nonsteepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the estimator previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

In this article for a finite typed random geometric graph we define the empirical locality distribution, which records the number of nodes of a given type linked to a given number of nodes of each type. We find large deviation principle (LDP) for the \emph{ empirical locality measure} given the empirical pair measure and the empirical type measure of the typed random geometric graphs. From this LDP, we derive large deviation principles for the \emph{degree measure and the proportion of detached nodes} in the classical Erd\H{o}s-R\'{e}nyi graph defined on $[0, 1]^d.$ This graphs have been suggested by (Canning and Penman, 2003) as a possible extension to the randomly typed random graphs.

In this article for a finite typed random geometric graph we define the\nempirical locality distribution, which records the number of nodes of a given\ntype linked to a given number of nodes of each type. We find large deviation\nprinciple (LDP) for the empirical locality measure given the empirical pair\nmeasure and the empirical type measure of the typed random geometric graphs.\nFrom this LDP, we derive large deviation principles for the degree measure and\nthe proportion of detached nodes in the classical Erdos-Renyi graph defined on\n[0, 1]^d. This graphs have been suggested by (Canning and Penman, 2003) as a\npossible extension to the randomly typed random graphs.\n

It is well known that symplectic methods have been rigorously shown to be superior to nonsymplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we attempt to study the superiority of stochastic symplectic methods by means of the large deviations principle. We propose the concept of asymptotical preservation of numerical methods for large deviations principles associated with the exact solutions of the general stochastic Hamiltonian systems. Considering that the linear stochastic oscillator is one of the typical stochastic Hamiltonian systems, we take it as the test equation in this paper to obtain precise results about the rate functions of large deviations principles for both exact and numerical solutions. Based on the Gärtner--Ellis theorem, we first study the large deviations principles of the mean position and the mean velocity for both the exact solution and its numerical approximations. Then, we prove that stochastic symplectic methods asymptotically preserve these two large deviations principles, but nonsymplectic ones do not. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the “hitting probability" of the mean position and mean velocity of the stochastic oscillator. Finally, numerical experiments are performed to show the superiority of stochastic symplectic methods in computing the large deviations rate functions. To the best of our knowledge, this is the first result about applying the large deviations principle to reveal the superiority of stochastic symplectic methods compared with nonsymplectic ones in the existing literature.