Large deviation principles for sequences of logarithmically weighted means

Sample path Large Deviations and optimal importance sampling for stochastic volatility models

Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions

By proving the continuity of multi-dimensional Skorokhod maps in a quasi-linearly discounted uniform norm on the doubly infinite time interval R, and strengthening know sample path large deviation principles for fractional Brownian motion to this topology, we obtain large deviation principles for the image of multi-dimensional fractional Brownian motions under Skorokhod maps as an immediate consequence of the contraction principle. As an application, we explicitly calculate large deviation decay rates for steady-state tail probabilities of certain queueing systems in multi-dimensional heavy traffic models driven by fractional Brownian motions.

We study a model of stochastic evolutionary game dynamics in which the probabilities that agents choose suboptimal actions are dependent on payoff consequences. We prove a sample path large deviation principle, characterizing the rate of decay of the probability that the sample path of the evolutionary process lies in a prespecified set as the population size approaches infinity. We use these results to describe excursion rates and stationary distribution asymptotics in settings where the mean dynamic admits a globally attracting state, and we compute these rates explicitly for the case of logit choice in potential games.

A magnetic body can be considered to consist of $n$ sites, where $n$ is large. The magnetic spins at these $n$ sites, whose sum is the total magnetization present in the body, can be modelled by a triangular array of random variables $(X^{(n)}_1,\ldots, X^{(n)}_n)$. Standard theory of physics would dictate that the joint distribution of the spins can be modelled by $dQ_n(\mathbf{x}) = z^{-1}_n \exp\lbrack -H_n(\mathbf{x})\rbrack\Pi dP(x_j)$, where $\mathbf{x} = (x_1,\ldots, x_n) \in \mathscr{R}^n$, where $H_n$ is the Hamiltonian, $z_n$ is a normalizing constant and $P$ is a probability measure on $\mathscr{R}$. For certain forms of the Hamiltonian $H_n$, Ellis and Newman (1978b) showed that under appropriate conditions on $P$, there exists an integer $r \geq 1$ such that $S_n/n^{1-1/2r}$ converges in distribution to a random variable. This limiting random variable is Gaussian if $r = 1$ and non-Gaussian if $r \geq 2$. In this article, utilizing the large deviation local limit theorems for arbitrary sequences of random variables of Chaganty and Sethuraman (1985), we obtain similar limit theorems for a wider class of Hamiltonians $H_n$, which are functions of moment generating functions of suitable random variables. We also present a number of examples to illustrate our theorems.

On the limiting behavior of randomly weighted partial sums

A law of the single logarithm for weighted sums of arrays applied to bootstrap model selection in regression

A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN

In this paper, a stochastic SIRS epidemic model with general awareness-induced and four independent Brownian Motions is established. We verify the global existence of a unique positive solution and find out the noise modified reproduction number R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> which is a sharp threshold for the dynamics: If R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> <; 1, the disease will die out; if R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> > 1, the disease persists and there exists a global asymptotically stable stationary distribution under parameter restrictive conditions. Numerical simulations are presented to illustrate the theoretical results.

Large Deviations of Realized Volatility

A 'chaos expansion' of the intersection local time functional of two independent Brownian motions in R d is given. The expansion is in terms of normal products of white noise (corresponding to multiple Wiener integrals). As a consequence of the local structure of the normal products, the kernel functions in the expansion are explicitly given and exhibit clearly the dimension dependent singularities of the local time functional. Their L p-properties are discussed. An important tool for deriving the chaos expansion is a computation of the 'S-transform' of the corresponding regularized intersection local times and a control about their singular limit.

The previous chapter considered in detail an example driven by a Brownian sheet, namely a stochastic reaction–diffusion equation. In this chapter we consider an application of one of the other formulations of infinite dimensional Brownian motion, which is the infinite sequence of independent one-dimensional Brownian motions. Such a collection will be used to define a general class of Brownian flows of diffeomorphisms [178], which are a special case of the stochastic flows of diffeomorphisms studied in [16, 25, 124, 178]. We will consider small noise asymptotics, prove the corresponding LDP, and then use it to give a Bayesian interpretation of an estimator used for image matching.

Let B 1,B 2,… be independent one-dimensional Brownian motions parameterized by the whole real line such that B i (0)=0 for every i≥1. We consider the nth iterated Brownian motion W n (t)=B n (B n−1(⋯(B 2(B 1(t)))⋯)). Although the sequence of processes (W n ) n≥1 does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W n converge to a random probability measure μ ∞. We then prove that μ ∞ almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W ∞ of independent Brownian motions. We also prove that the collection of random variables (W ∞(t),t∈ℝ∖{0}) is exchangeable with directing measure μ ∞.

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.

Annals of operations research (vol. 28 - 1991) Markov decision processes — Part I Annals of operations research (vol. 29 - 1991) Markov decision processes — Part II: O. Hernandez-Lerma and J.B. Lasserre (eds.) Baltzer AG, Basel, 1991, 649 pages, ISSN 0254 5330