Properties of a functional of trajectories which arises in studying the probabilities of large deviations of random walks

The deviation functional (or integral) describes the logarithmic asymptotics of the probabilities of large deviations of trajectories of the random walks generated by the sums of random variables (vectors) (see [1, 2] for instance). In this article we define it on a broader function space than previously and under weaker assumptions on the distributions of jumps of the random walk. The deviation integral turns out the Darboux integral ∫ F(t, u) of a semiadditive interval function F(t, u) of a particular form. We study the properties of the deviation integral and use the results elsewhere in [3] to prove some generalizations of the large deviation principle established previously under rather restrictive assumptions.     

A multivariate empirical characteristic function test of independence with normal marginals

This paper proposes a semi-parametric test of independence (or serial independence) between marginal vectors each of which is normally distributed but without assuming the joint normality of these marginal vectors. The test statistic is a Cramér–von Mises functional of a process defined from the empirical characteristic function. This process is defined similarly as the process of Ghoudi et al. [J. Multivariate Anal. 79 (2001) 191] built from the empirical distribution function and used to test for independence between univariate marginal variables. The test statistic can be represented as a V-statistic. It is consistent to detect any form of dependence. The weak convergence of the process is derived. The asymptotic distribution of the Cramér–von Mises functionals is approximated by the Cornish–Fisher expansion using a recursive formula for cumulants and inversion of the characteristic function with numerical evaluation of the eigenvalues. The test statistic is finally compared with Wilks statistic for testing the parametric hypothesis of independence in the one-way MANOVA model with random effects.     

One‐Bit Compressed Sensing by Linear Programming

We give the first computationally tractable and almost optimal solution to the problem of one‐bit compressed sensing, showing how to accurately recover an s‐sparse vector \input amssym $x \in {\Bbb R}^n$ from the signs of $O(s \log^2(n/s))$ random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our result is universal in the sense that with high probability, one measurement scheme will successfully recover all sparse vectors simultaneously. The argument is based on solving an equivalent geometric problem on random hyperplane tessellations.     

A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums , whereX _n,n⩾1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, andN(t), t⩾0 is a process of non-negative integer-valued random variables, independent ofX _n,n⩾1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what reasonable condition can be given onN(t), t⩾0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.     

Extremes of Lévy driven mixed MA processes with convolution equivalent distributions

We investigate the extremal behavior of stationary mixed MA processes for t ≥ 0, where f is a deterministic function and Λ is an infinitely divisible and independently scattered random measure. Particular examples of mixed MA processes are superpositions of Ornstein-Uhlenbeck processes, applied as stochastic volatility models in Barndorff-Nielsen and Shephard (2001a). We assume that the finite dimensional distributions of Λ are in the class of convolution equivalent tails and in the maximum domain of attraction of the Gumbel distribution. On the one hand, we compute the tail behavior of Y(0) and sup _{t ∈ [0,1]} Y(t). On the other hand, we study the extremes of Y by means of marked point processes based on maxima of Y in random intervals. A complementary result guarantees the convergence of the running maxima of Y to the Gumbel distribution. Financial support from the Deutsche Forschungsgemeinschaft through a research grant is gratefully acknowledged.     

The structure of a Brownian bubble

Summary We examine local geometric properties of level sets of the Brownian sheet, and in particular, we identify the asymptotic distribution of the area of sets which correspond to excursions of the sheet high above a given level in the neighborhood of a particular random point. It is equal to the area of certain individual connected components of the random set {(s, t):B(t)>b(s)}, whereB is a standard Brownian motion andb is (essentially) a Bessel process of dimension 3. This limit distribution is studied and, in particular, explicit formulas are given for the probability that a point belongs to a specific connected component, and for the expected area of a component given the height of the excursion ofB(t)-b(s) in this component. These formulas are evaluated numerically and compared with the results from direct simulations ofB andb.The research of this author was partially supported by grants DMS-9103962 from the National Science Foundation and DAAL03-92-6-0323 from the Army Research Office     

Banach-Mazur Distances and Projections on Random Subgaussian Polytopes

We consider polytopes in that are generated by N vectors in whose coordinates are independent subgaussian random variables. (A particular case of such polytopes are symmetric random polytopes generated by N independent vertices of the unit cube.) We show that for a random pair of such polytopes the Banach-Mazur distance between them is essentially of a maximal order n. This result is an analogue of the well-known Gluskin's result for spherical vectors. We also study the norms of projections on such polytopes and prove an analogue of Gluskin's and Szarek's results on basis constants. The proofs are based on a version of "small ball" estimates for linear images of random subgaussian vectors.     

Martin boundaries for some space-time Markov processes

Summary One considers a simple exclusion particle jump process on , where the underlying one particle motion is a degenerate random walk that moves only to the right. One starts with the configuration in which the left halfline is completely occupied and the right one free. It is shown that the number of particles at time t between site [u t] and [v t], divided by t, converges a.s. to , where f might be called the density profile. It is explicitely determined and shown to be an affine function. Secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u, converges weakly to the Bernoulli measure with density f(u), as the time tends to infinity.This work has been supported by the Deutsche Forschungsgemeinschaft     

Comportement asymptotique des fonctions de répartition perturbées pour des processus non stationnaires et absolument réguliers

The object is to study the asymptotic normality of the statistics associated to the perturbed empirical distribution function via the slow convergence of multivariate U-statistic. We extend the results of Sun (1993) from the case of identically distributed absolutely regular random variables to the case of nonstationary absolutely regular random vectors. To cite this article: M. Harel, E. Elharfaoui, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Convex geometry of max-stable distributions

It is shown that max-stable random vectors in [0, ∞ ) ^d with unit Fréchet marginals are in one to one correspondence with convex sets K in [0, ∞ ) ^d called max-zonoids. The max-zonoids can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or, alternatively, as the selection expectation of a random cross-polytope whose distribution is controlled by the spectral measure of the max-stable random vector. Furthermore, the cumulative distribution function P ξ ≤ x of a max-stable random vector ξ with unit Fréchet marginals is determined by the norm of the inverse to x, where all possible norms are given by the support functions of (normalised) max-zonoids. As an application, geometrical interpretations of a number of well-known concepts from the theory of multivariate extreme values and copulas are provided.      

风险模型中重尾随机变量和的若干大偏差结果

进一步研究随机变量部分和与随机和的大偏差,其中S(n)=∑ni=1Xi,S(t)=∑N(t)i=1Xi(t>0).{Xn,n≥1}是一个独立同分布的随机变量(未必是非负的)序列具有共同的分布F(定义于R上)和有限期望μ=EX1.{N(t),t≥0}是一个非负的整数值的随机变量的更新计数过程且与{Xn,n≥1}相互独立.本文在假定F∈C条件下,进一步推广并改进了由Klüppelberg等和Kaiw等人给出的一些大偏差结果.这些结果可应用到某些金融保险方面的一些特定的问题中去.     

Generalized skew-elliptical distributions and their quadratic forms

This paper introduces generalized skew-elliptical distributions (GSE), which include the multivariate skew-normal, skew-t, skew-Cauchy, and skew-elliptical distributions as special cases. GSE are weighted elliptical distributions but the distribution of any even function in GSE random vectors does not depend on the weight function. In particular, this holds for quadratic forms in GSE random vectors. This property is beneficial for inference from non-random samples. We illustrate the latter point on a data set of Australian athletes.     

Optimal estimates of random vectors using information criteria

Information criteria are applied for estimation of random vectors. Normal random vectors and random vectors with an unknown distribution are considered. Both linear estimates and estimates represented by a measurable function of observations are derived.Translated from Vychislitel'naya i Prildadnaya Matematika, No. 62, pp. 105–113, 1987.     

PERIODIC SOLUTIONS OF LINEAR NEUTRAL INTEGRO-DIFFERENTIAL EQUATIONS

Consider the linear neutral FDEd/dt[x(t) Ax(t - r)] = R [dL(s)]x(t s) f(t)where x and / are ra-dimensional vectors; A is an n x n constant matrix and L(s) is an n x n matrix function with bounded total variation. Some necessary and sufficient conditions are given which guarantee the existence and uniqueness of periodic solutions to the above equation.     

Large Deviations for Sums of Independent Heavy-Tailed Random Variables

We obtain precise large deviations for heavy-tailed random sums , of independent random variables. are nonnegative integer-valued random variables independent of r.v. (X _i )i N with distribution functions F _i. We assume that the average of right tails of distribution functions F _i is equivalent to some distribution function with regularly varying tail. An example with the Pareto law as the limit function is given.     

Asymptotic expansions in the central limit theorem for compound and Markov processes

Summary For independent identically distributed bivariate random vectors (X ₁, Y ₁), (X ₂, Y ₂), ... and for large t the distribution of X ₁ +...+ X _N(t) is approximated by asymptotic expansions. Here N(t) is the counting process with lifetimes Y ₁, Y ₂,.... Similar expansions are derived for multivariate X ₁. Furthermore, local asymptotic expansions are valid for the distribution of f(X ₁)+ ...+ f(X _N ) when N is large and nonrandom, and X _i , i=1, 2,..., is a discrete strongly mixing Markov chain.     

On the existence of ordered couplings of random sets — with applications

Letψ andϕ be two given random closed sets in a locally compact second countable topological spaceS. (They need not be based on the same probability space.) The main result gives necessary and sufficient conditions on the distributions ofψ andϕ, for the existence of two random closed sets and , based on the same probability space and such that their distributions coincide with those ofψ andϕ, resp., and a.s. This coupling result tells us in particular when a probability distribution onS is selectionable w.r.t. (the distribution of) a random closed set. An existence result for realizable thinnings of a simple point process is obtained by specializing it to supports of random measures. The coupling result is extended to random variables in a countably based continuous poset. As examples we mention various kinds of random capacities — in particular random measures — and random compact (saturated) sets. Moreover, the extended result tells us when a probability distribution onS is selectionable w.r.t. the distribution of a random compact (saturated) set.     

Limit theorems for BSDE with local time applications to non-linear PDE

Given a d -dimensional Wiener process W , with its natural filtration F t , a F T -measurable random variable ξ in R , a bounded measure x on R , and an adapted process ( s , y , z ) M h ( s , y , z ), we consider the following BSDE: Y t = ξ + Z t T h ( s , Y s , Z s ) d s + Z R ( L T a ( Y ) m L t a ( Y )) x (d a ) m Z t T Z s d W s for 0 h t h T . Here L t a ( Y ) stands for the local time of Y at level a . For h =0, we establish the existence and the uniqueness of the processes ( Y , Z ), and if h is continuous with linear growth we establish the existence of a solution. We prove limit theorems for solutions of backward stochastic differential equations of the above form. Those limit theorems permit us to deduce that any solution of that equation is the limit, in a strong sense, of a sequence of semi-martingales, which are solutions of ordinary BSDEs of the form Y t = ξ + Z t T f ( Y s ) Z s 2 d s m Z t T Z s d W s . A comparison theorem for BSDEs involving measures is discussed. As an application we obtain, with the help of the connection between BSDE and PDE, some corresponding limit theorems for a class of singular non-linear PDEs and a new probabilistic proof of the comparison theorem for PDEs.     

Bounds on the value of information in uncertain decision problems

The cost of obtaining good information regarding the various probability distributions needed for the solution of most stochastic decision problems is considerable. It is important to consider questions such as: (1) what minimal amounts of information are sufficient to determine optimal decision rules; (2) what is the value of obtaining knowledge of the actual realization of the random vectors; and (3) what is the value of obtaining some partial information regarding the actual realization of the random vectors. This paper is primarily concerned with questions two and three when the decision maker has an a priori knowledge of the joint distribution function of the random variables. Some remarks are made regarding results along the lines of question one. Mention is made of assumptions sufficient so that knowledge of means, or of means, variances, co-variances and n-moments are sufficient for the calculation of optimal decision rules. The analysis of the second question leads to the development of bounds on the value of perfect information. For multiperiod problems it is important to consider when the perfect information is available. Jensen's inequality is the key tool of the analysis. The calculation of the bounds requires the solution of nonlinear programs and the numerical evaluation of certain functions. Generally speaking, tighter bounds may be obtained only at the expense of additional information and computational complexity. Hence, one may wish to compute some simple bounds to decide upon the advisability of obtaining more information. For the analysis of the value of partial information it is convenient to introduce the notion of a signal. Each signal represents the receipt of certain information, and these signals are drawn from a given probability distribution. When a signal is received, it alters the decision maker's perception of the probability distributions inherent in his decision problem. The choice between different information structures must then take into account these probability distributions as well as the decision maker's preference function. A hierarchy of bounds may be determined for partial information evaluation utilizing the tools of the multiperiod perfect information case. However, the calculation of these bounds is generally considerably more dicult than the calculation of similar boulids in the perfect information case. Most of the analysis is directed towards problems in which the decision maker has a linear utility function over profits, costs or some other numerical variable. However, some of the bounds generalize to the case when the utility function is strictly increasing and concave.     

Strong Convergence of the Kernel Estimates of Nonparametric Regression Functions

Let (X, Y), (X_1, Y_1),\cdots, (X_n, Y_n) be i. i. d. random vectors taking values in R_d\times R with E(|Y|)<\infinity, To estimate the regression function m(x)=E(Y|X=x), we use the kernel estimate $m_n(x)=[\sum\limits_{i = 1}^n {K(\frac{{{X_i} - x}}{{{h_n}}}){Y_i}/} \sum\limits_{i = 1}^n {K(\frac{{{X_j} - x}}{{{h_n}}})} \]$ where K(x) is a kernel function and h_n a window width. In this paper, we establish the strong consistency of m_n(x) when E(|Y|^p)<\infinity for some p>l or E{exp(t|Y|^\lambda)}<\infinity for some \lambda>0 and t>0. It is remakable that other conditions imposed here are independent of the distribution of (X, Y).