Large Deviation Result Research Articles

Large deviation results for generalized compound negative binomial risk models

In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {Xn; n ≥ 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean µ ∈ R+, {N(n); n ≥ 0} be a sequence of negative binomial distributed random variables with a parameter p ∈ (0, 1), n ≥ 0, let {M(n); n ≥ 0} be a Poisson process with intensity λ > 0. Suppose {N(n); n ≥ 0}, {Xn; n ≥ 1} and {M(n); n ≥ 0} are mutually independent. Write S(n) = \( \sum\limits_{i = 1}^{N(n)} {X_i - cM(n)} \). Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.

Moderate deviations for Poisson–Dirichlet distribution

The Poisson--Dirichlet distribution arises in many different areas. The parameter $\theta$ in the distribution is the scaled mutation rate of a population in the context of population genetics. The limiting case of $\theta$ approaching infinity is practically motivated and has led to new, interesting mathematical structures. Laws of large numbers, fluctuation theorems and large-deviation results have been established. In this paper, moderate-deviation principles are established for the Poisson--Dirichlet distribution, the GEM distribution, the homozygosity, and the Dirichlet process when the parameter $\theta$ approaches infinity. These results, combined with earlier work, not only provide a relatively complete picture of the asymptotic behavior of the Poisson--Dirichlet distribution for large $\theta$, but also lead to a better understanding of the large deviation problem associated with the scaled homozygosity. They also reveal some new structures that are not observed in existing large-deviation results.

Large deviation estimates of the crossing probability for pinned Gaussian processes

The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay innfixed points atnfixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motions andm-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time of the unpinned process.

Large deviation estimates of the crossing probability for pinned Gaussian processes

The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motions and m-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time of the unpinned process.

Precise large deviation results for the total claim amount under subexponential claim sizes

The paper deals with the renewal risk model. A precise large deviation result in the case of subexponential claim sizes is proved. As a special case, the example of Pareto distributed claim sizes and inter-occurrence times is investigated.

The general principle for precise large deviations of heavy-tailed random sums

Let { X n ; n ⩾ 1 } be an arbitrary sequence of heavy-tailed random variables, independent of a process of nonnegative, integer-valued rv's { N ( t ) ; t ⩾ 0 } . In this paper, we present a necessary and a sufficient condition on { N ( t ) ; t ⩾ 0 } under which the precise large deviation result for nonrandom sum S n = ∑ i = 1 n X i can be extended to the random sum S N ( t ) . The results we obtain not only generalize but also improve some related classical ones.

Large Deviation Results for Periodic Points of a Rational Map

In this note we show a large deviation result for the periodic points of a hyperbolic rational map on the Riemann sphere. This extends the well known equidistribution result of Lyubich in this setting. We also consider convergence results for more general weighted averages of orbital measures with respect to Hölder continuous functions.

Maxima of the cells of an equiprobable multinomial

Consider a sequence of multinomial random vectors with increasing number of equiprobable cells. We show that if the number of trials increases fast enough, the sequence of maxima of the cells after a suitable centering and scaling converges to the Gumbel distribution. While results are available for maxima of triangular arrays of independent random variables with certain types of distribution, such results in a dependent setup is new. We also prove that the maxima of a triangular sequence of appropriate Binomial random variables have the same limit distribution. An auxiliary large deviation result for multinomial distribution with increasing number of equiprobable cells may also be of independent interest.

Cluster Sets of Self-Normalized Sums

We determine the cluster sets of certain self-normalized sums of i.i.d. random variables. In the process, we obtain a refined large deviation result for sums in the domain of attraction of a stable law.

Dynamic quorum policy for maximizing throughput in limited information multiparty MAC

In multiparty MAC, a sender needs to transmit each packet to a set of receivers within its transmission range. Bandwidth efficiency of wireless multiparty MAC can be improved substantially by exploiting the fact that several receivers can be reached at the MAC layer by a single transmission. Multiparty communication, however, requires new design paradigms since systematic design techniques that have been used effectively in unicast and wireline multicast do not apply. For example, a transmission policy that maximizes the stability region of the network need not maximize the network throughput. Therefore, the objective is to design a policy that maximizes the system throughput subject to maintaining stability. We present a sufficient condition that can be used to establish the throughput optimality of a stable transmission policy. We subsequently design a distributed adaptive stable policy that allows a sender to decide when to transmit using simple computations. The computations are based only on limited information about current transmissions in the sender's neighborhood. Even though the proposed policy does not use any network statistics, it attains the same throughput as an optimal offline stable policy that uses in its decision process past, present, and even future network states. We prove the throughpOut optimality of this policy using the sufficient condition and the large deviation results. We present a MAC protocol for acquiring the local information necessary for executing this policy, and implement it in ns-2. The performance evaluations demonstrate that the optimal policy significantly outperforms the existing multiparty schemes in ad hoc networks.

Large deviations for trapped interacting Brownian particles and paths

We introduce two probabilistic models for N interacting Brownian motions moving in a trap in ℝd under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyze both models in the limit of diverging time with fixed number N of Brownian motions. In particular, we prove large deviations principles for the normalized occupation measures. The minimizers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of N interacting trapped particles. More precisely, in the particle-repellency model, the minimizer is its ground state, and in the path-repellency model, the minimizers are its ground product-states. In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model. This study is a contribution to the search for a mathematical formulation of the quantum system of N trapped interacting bosons as a model for Bose–Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behavior of the ground state in terms of the well-known Gross–Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behavior of the ground product-states is also described by the Gross–Pitaevskii formula, however, with the scattering length of the pair potential replaced by its integral.

On approximate pattern matching for a class of Gibbs random fields

We prove an exponential approximation for the law of approximate occurrence of typical patterns for a class of Gibssian sources on the lattice ℤd, d≥2. From this result, we deduce a law of large numbers and a large deviation result for the waiting time of distorted patterns.

Yule process sample path asymptotics

This paper presents two results on sample paths for the Yule process: one fluid limit theorem and one sample path large deviation result. The main interest is to understand the way large deviation occurs in the case of non-homogeneous processes. There are indeed two new phenomena. First there is no ``typical'' speed of large deviation. Second, the large deviation event is concentrated on a finite interval of time.

Bahadur exact slopes of some tests for spectral densities

The large deviation result is proved for two functionals of the empirical spectral process in zero-mean Gaussian stationary processes. As a statistical application, we deal with the Bahadur asymptotic efficiencies of two statistics for testing H: f 1 = f (specified), which are spectral analogue to the Kolmogorov-Smirnov (KS) and Kuiper statistics for testing hypothesis about distribution function in the iid setting. It is shown that the Kuiper type statistic is superior to the KS type statistic in terms of the Bahadur exact slope. We also discuss the a( ≥ 2)-sample problem. Especially, for the two-sample problem, we investigate the Bahadur asymptotic efficiencies of several statistics for testing not only the goodness-of-fit hypothesis H 1: f 1 = f 2 = f (specified) but also the homogeneity hypothesis H 2: f 1 = f 2 (unspecified).

A large deviation result for aggregate claims with dependent claim occurrences

In this paper we study precise large deviations of a compound sum of claims, in which the claims arrive in groups and the claim numbers in the groups may follow a certain negative dependence structure. We try to build a platform both for the classical large deviation theory and for the modern stochastic ordering theory.

Large deviations for a class of nonhomogeneous Markov chains

Large deviation results are given for a class of perturbed nonhomogeneous Markov chains on finite state space which formally includes some stochastic optimization algorithms. Specifically, let {Pn} be a sequence of transition matrices on a finite state space which converge to a limit transition matrix P. Let {Xn} be the associated nonhomogeneous Markov chain where Pn controls movement from time n−1 to n. The main statements are a large deviation principle and bounds for additive functionals of the nonhomogeneous process under some regularity conditions. In particular, when P is reducible, three regimes that depend on the decay of certain “connection” Pn probabilities are identified. Roughly, if the decay is too slow, too fast or in an intermediate range, the large deviation behavior is trivial, the same as the time-homogeneous chain run with P or nontrivial and involving the decay rates. Examples of anomalous behaviors are also given when the approach Pn→P is irregular. Results in the intermediate regime apply to geometrically fast running optimizations, and to some issues in glassy physics.

A large deviation result for the subcritical Bernoulli percolation

Nous considerons la percolation de Bernoulli dans les dimensions superieures ou egales a deux. Si C est le cluster d'aretes ouvertes contenant l'origine, nous prouvons que la loi de C/N satisfait un principe de grandes deviations par rapport a la metrique de Hausdorff.

Large deviations for supercritical multitype branching processes

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + x α F *[i](x) + o(x α ) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x −β/(1−β) G*[i](x) + o (x −β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = x δ/(δ−1) H *[i](x) + o(x δ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F *, G * and H * are multiplicatively periodic functions.

Large deviations for supercritical multitype branching processes

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + xαF*[i](x) + o(xα) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x−β/(1−β) G*[i](x) + o (x−β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = xδ/(δ−1)H*[i](x) + o(xδ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F*, G* and H* are multiplicatively periodic functions.

Large deviations of [formula omitted]-error of empirical measures on partitions

We obtain a Chernoff-type large deviation result for the L 1 -error of empirical measure on partitions. As a corollary, large deviation asymptotics for the L 1 -error of the histogram density estimator is derived. Applications to Bahadur efficiencies of goodness-of-fit tests and Bahadur exact slopes of these statistics are discussed.