Large convex cones in hypercubes

A family of subsets of [n] is positive linear combination free if the characteristic vector of neither member is the positive linear combination of the characteristic vectors of some other ones. We construct a positive linear combination free family which contains (1-o(1))2ⁿ subsets of [n] and we give tight bounds on the o(1)2ⁿ term. The problem was posed by Ahlswede and Khachatrian [Cone dependence—a basic combinatorial concept, Preprint 00-117, Diskrete Strukturen in der Mathematik SFB 343, Universität Bielefeld, 2000] and the result has geometric consequences.     

Critical angles in random polyhedral cones

This work concerns the angular structure of random polyhedral cones generated by p stochastically independent vectors uniformly distributed on the unit sphere of Rn. We comment on the expected number of critical angles and the mathematical expectation of the extremal angles.     

Large deviations and phase transition for random walks in random nonnegative potentials

We establish large deviation principles and phase transition results for both quenched and annealed settings of nearest-neighbor random walks with constant drift in random nonnegative potentials on Z^d${\mathbb{Z}}^d$. We complement the analysis of M.P.W. Zerner [Directional decay of the Green’s function for a random nonnegative potential on Z^d${\mathbb{Z}}^d$, Ann. Appl. Probab. 8 (1996) 246–280], where a shape theorem on the Lyapunov functions and a large deviation principle in absence of the drift are achieved for the quenched setting.     

Tails of random sums of a heavy-tailed number of light-tailed terms

The tail of the distribution of a sum of a random number of independent and identically distributed nonnegative random variables depends on the tails of the number of terms and of the terms themselves. This situation is of interest in the collective risk model, where the total claim size in a portfolio is the sum of a random number of claims. If the tail of the claim number is heavier than the tail of the claim sizes, then under certain conditions the tail of the total claim size does not change asymptotically if the individual claim sizes are replaced by their expectations. The conditions allow the claim number distribution to be of consistent variation or to be in the domain of attraction of a Gumbel distribution with a mean excess function that grows to infinity sufficiently fast. Moreover, the claim number is not necessarily required to be independent of the claim sizes.     

Large deviations for Brownian motion in a random scenery

We prove large deviations principles in large time, for the Brownian occupation time in random scenery . The random field is constant on the elements of a partition of ^d into unit cubes. These random constants, say consist of i.i.d. bounded variables, independent of the Brownian motion {B_s,s0}. This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in ``quenched' and ``annealed' settings.Mathematics Subject Classification (2000):60F10, 60J55, 60K37     

Large deviations of realized volatility

This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency financial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a fixed time interval increases to infinity. Our large deviation result can be used to evaluate tail probabilities of the realized volatility. We also derive a moderate deviation rate function for a standardized realized volatility statistic. The moderate deviation result is useful for assessing the validity of normal approximations based on the central limit theorem. In particular, it clarifies that there exists a trade-off between the accuracy of the normal approximations and the path regularity of an underlying volatility process. Our large and moderate deviation results complement the existing asymptotic theory on high frequency data. In addition, the paper contributes to the literature of large deviation theory in that the theory is extended to a high frequency data environment.     

Jensen's inequality for random elements in metric spaces and some applications

Jensen's inequality is extended to metric spaces endowed with a convex combination operation. Applications include a dominated convergence theorem for both random elements and random sets, a monotone convergence theorem for random sets, and other results on set-valued expectations in metric spaces and on random probability measures.     

Bifractional inequalities and convex cones

In this paper we give a systematic study of a class of linear inequalities related to convex cones in linear spaces. In particular, Chebyshev and Andersson type inequalities are discussed. Some classical and new inequalities are derived from the results.     

Efficient simulation and conditional functional limit theorems for ruinous heavy-tailed random walks

The contribution of this paper is to introduce change of measure based techniques for the rare-event analysis of heavy-tailed random walks. Our changes of measures are parameterized by a family of distributions admitting a mixture form. We exploit our methodology to achieve two types of results. First, we construct Monte Carlo estimators that are strongly efficient (i.e. have bounded relative mean squared error as the event of interest becomes rare). These estimators are used to estimate both rare-event probabilities of interest and associated conditional expectations. We emphasize that our techniques allow us to control the expected termination time of the Monte Carlo algorithm even if the conditional expected stopping time (under the original distribution) given the event of interest is infinity–a situation that sometimes occurs in heavy-tailed settings. Second, the mixture family serves as a good Markovian approximation (in total variation) of the conditional distribution of the whole process given the rare event of interest. The convenient form of the mixture family allows us to obtain functional conditional central limit theorems that extend classical results in the literature.     

Large deviations for truncated heavy-tailed random variables: A boundary case

This paper investigates the decay rate of the probability that the row sum of a triangular array of truncated heavy tailed random variables is larger than an integer (k) times the truncating threshold, as both - the number of summands and the threshold go to infinity. The method of attack for this problem is significantly different from the one where k is not an integer, and requires much sharper estimates.     

Asymptotic analysis of risk quantities conditional on ruin for multidimensional heavy-tailed random walks

In this paper we consider a multidimensional renewal risk model with regularly varying claims. This model may be used to describe the surplus of an insurance company possessing several lines of business where a large claim possibly puts multiple lines in a risky condition. Conditional on the occurrence of ruin, we develop asymptotic approximations for the average accumulated number of claims leading the process to a rare set, and the expected total amount of shortfalls to this set in finite and infinite horizons. Furthermore, for the continuous time case, asymptotic results regarding the total occupation time of the process in a rare set and time-integrated amount of shortfalls to a rare set are obtained.     

LARGE DEVIATIONS FOR FIELDS WITH STATIONARY INDEPENDENT INCREMENTS

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Large deviations for stochastic generalized porous media equations

The large deviation principle is established for the distributions of a class of generalized stochastic porous media equations for both small noise and short time.     

Projective and inductive limits in locally convex cones

We define and study the projective and inductive limit notions for locally convex cones. We use convex quasiuniform structure method for this purpose. Also we study the barreledness in the locally convex cones and introduce the notion upper-barreled cones and prove that the inductive limit of upper-barreled cones is upper-barreled.     

LARGE DEVIATIONS FOR INFINITE DIMENSIONAL AND REVERSIBLE REACTION－DIFFUSION PROCESSES

ＬＡＲＧＥＤＥＶＩＡＴＩＯＮＳＦＯＲＩＮＦＩＮＩＴＥＤＩＭＥＮＳＩＯＮＡＬＡＮＤＲＥＶＥＲＳＩＢＬＥＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＰＲＯＣＥＳＳＥＳＣＨＥＮＪＩＮＷＥＮ（陈金文）（ＤｅｐｅｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，Ｔ...     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

Large deviations bounds for estimating conditional value-at-risk

In this paper, we prove an exponential rate of convergence result for a common estimator of conditional value-at-risk for bounded random variables. The bound on optimistic deviations is tighter while the bound on pessimistic deviations is more general and applies to a broader class of convex risk measures.     

特殊重尾随机变量随机和的大偏差

文献[1]对于一些经典重尾随机变量的随机和大偏差作了有意义的讨论,本文则讨论了另外一些同样有用的重尾随机和的大偏差.     

Large deviations for diffusion processes in duals of nuclear spaces

Motivated by applications to neurophysiological problems, various authors have studied diffusion processes in duals of countably Hilbertian nuclear spaces governed by stochastic differential equations. In these models the diffusion coefficients describe the random stimuli received by spatially extended neurons. In this paper we present a large deviation principle for such processes when the diffusion terms tend to zero in terms of a small parameter. The lower bounds are established by making use of the Girsanov formula in abstract Wiener space. The upper bounds are obtained by Gaussian approximation of the diffusion processes and by taking advantage of the nuclear structure of the state space to pass from compact sets to closed sets.This research was partially supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620-92-J-0154 and the Army Research Office Grant No. DAAL03-92-G-0008.