Local Large Deviation Research Articles

Large deviations for processes with discontinuous statistics

This paper is devoted to the problem of sample path large deviations for the Markov processes on R_+^N having a constant but different transition mechanism on each boundary set {x:x_i=0 for i\notin\Lambda, x_i>0 for i\in\Lambda}. The global sample path large deviation principle and an integral representation of the rate function are derived from local large deviation estimates. Our results complete the proof of Dupuis and Ellis of the sample path large deviation principle for Markov processes describing a general class of queueing networks.

An explicit local uniform large deviation bound for Brownian bridges

By comparing curve length in a manifold and a standard sphere, we prove a local uniform bound for the exponent in the Large Deviation formula that describes the concentration of Brownian bridges to geodesics.

Local limit theory and large deviations for supercritical Branching processes

In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Zn:n≥1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Zn, that is, the behavior of P(Zn=vn) as vn↗∞, and use this to study conditional large deviations of {YZn:n≥1}, where Yn satisfies an LDP, particularly of {Zn−1Zn+1:n≥1} conditioned on Zn≥vn.

Large Deviations for the Super-Brownian Motion with Super-Brownian Immigration

Local large deviation principles are established in dimensions d≥3 for the super Brownian motion with random immigration X ϱ t , where the immigration rate is governed by the trajectory of another super-Brownian motion ϱ. The speed function is t for d≥4 and t1/2 for d=3, compared with the existing results, the interesting phenomenon happened in d=4 with speed t (although only the upper large deviation bound is derived here) is just because the structure of this new model: the random immigration “smooth” the critical dimension in some sense. The rate function are characterized by an evolution equation.

On the large deviations approximation for the stationary distribution of skip-free regulated queueing networks

Large deviations papers like that of Ignatyuk et al. [Russ. Math. Surv. 49 (1994) 41–99] have shown that asymptotically, the stationary distribution of homogeneous regulated networks is of the form P{X(t)≈n}≈K e −αn with the coefficient α being different in various “boundary influence domains” and also depending on some of these domains on n. In this paper, we focus on the case of constant exponents α and on a subclass of networks we call “strongly skip-free” (which includes all Jackson and all two-dimensional skip-free networks). We conjecture that an asymptotic exponent α is constant iff it corresponds to a large deviations escape path which progresses gradually (from the origin to the interior) through boundary facets whose dimension always increases by one. Solving the corresponding large deviations problem for our subclass of networks leads to a family of “local large deviation systems” (LLDSs) (for the constant exponents), which are expressed entirely in terms of the cumulant generating function of the network. In this paper, we show that at least for “strongly skip-free” Markovian networks with independent transition processes, the LLDS is closely related to some “local boundary equilibrium systems” (LESs) obtained by retaining from the equilibrium equations only those valid in neighborhoods of the boundary. Since asymptotic results require typically only that the cumulant generating function is well-defined over an appropriate domain, it is natural to conjecture that these LLDSs will provide the asymptotic constant exponents regardless of any distributional assumptions made on the network. Finally, we outline a practical recipe for combining the local approximations to produce a global large deviations approximation P{X t≈n}≈∑ jK j e −α (j)n , with the coefficients K j determined numerically.

Large Deviation Principle for Poisson Random Variables and Young Diagrams

We prove the local large deviation principle (LLDP) for the shape of a random Young diagram with a specific distribution using the large deviation principle for a sequence of Poisson random variables.

One-sided local large deviation and renewal theorems in the case of infinite mean

If {S n ,n≧0} is an integer-valued random walk such that S n /a n converges in distribution to a stable law of index α∈ (0,1) as n→∞, then Gnedenko’s local limit theorem provides a useful estimate for P{S n =r} for values of r such that r/a n is bounded. The main point of this paper is to show that, under certain circumstances, there is another estimate which is valid when r/a n → +∞, in other words to establish a large deviation local limit theorem. We also give an asymptotic bound for P{S n =r} which is valid under weaker assumptions. This last result is then used in establishing some local versions of generalized renewal theorems.

The large deviation principle for a general class of queueing systems. I

We prove the existence of a rate function and the validity of the large deviation principle for a general class of jump Markov processes that model queueing systems. A key step in the proof is a local large deviation principle for tubes centered at a class of piecewise linear, continuous paths mapping [0,1] into [ 0 , 1 ] [0,1] . In order to prove certain large deviation limits, we represent the large deviation probabilities as the minimal cost functions of associated stochastic optimal control problems and use a subadditivity—type argument. We give a characterization of the rate function that can be used either to evaluate it explicitly in the cases where this is possible or to compute it numerically in the cases where an explicit evaluation is not possible.

Uniform Convergence of Martingales in the Branching Random Walk

In a discrete-time supercritical branching random walk, let $Z^{(n)}$ be the point process formed by the $n$th generation. Let $m(\lambda)$ be the Laplace transform of the intensity measure of $Z^{(1)}$. Then $W^{(n)}(\lambda) = \int e^{-\lambda x}Z^{(n)}(dx)/m(\lambda)^n$, which is the Laplace transform of $Z^{(n)}$ normalized by its expected value, forms a martingale for any $\lambda$ with $|m(\lambda)|$ finite but nonzero. The convergence of these martingales uniformly in $\lambda$, for $\lambda$ lying in a suitable set, is the first main result of this paper. This will imply that, on that set, the martingale limit $W(\lambda)$ is actually an analytic function of $\lambda$. The uniform convergence results are used to obtain extensions of known results on the growth of $Z^{(n)}(nc + D)$ with $n$, for bounded intervals $D$ and fixed $c$. This forms the second part of the paper, where local large deviation results for $Z^{(n)}$ which are uniform in $c$ are considered. Finally, similar results, both on martingale convergence and uniform local large deviations, are also obtained for continuous-time models including branching Brownian motion.

Non-Stationary Chaos Revisited from Large Deviation Theory

The long time tail of the modified Bernoulli map is discussed in terms of the large deviation theory. The transition from stationary regime to non-stationary regime is char· acterized by the asymptotic scaling form of the entropy function. The concept of local large deviation property is proposed in understanding the critical regime of the transition. The multi-ergodic feature observed in hamiltonian chaos is briefly considered from the viewpoint of the multiple phase coexistence with different local large deviation properties.