General max-stable laws

We review the work on max-stable laws and their max domains of attraction introduced by Pancheva (Lect Notes Math 1155:284–309, 1984). We introduce the concept of general max domain of strict attraction of the general max-stable laws, a subclass of the general max domain of attraction and prove new results. Some interesting examples also are discussed.      

Norming operators for generalized domains of attraction

A sequence of independent and identically distributed random vectorsX _n on ^k is said to belong to the generalized domain of attraction of a nondegenerate random vectorY on ^k provided that there exist linear operatorsA _n on ^k and nonrandom constantsb _n ^k such that the centered and normalized partial sumsA _n (X ₁++X _n –b _n converge in distribution toY. In this paper we show that the sequence of norming operatorsA _n can always be chosen to vary regularly.Partially supported by NSF Grant DMS-91-03131 at Albion College.     

Domains of attraction of nonnormal operator-stable laws

A sequence of independent, identically distributed random vectors X₁, X₂, X₃,… is said to belong to the domain of attraction of a random vector Y is there exist linear operators A_n and constant vectors b_n such that A_n(X₁,…, X_n)+b_n converges in distribution to Y. We present a simple, necessary, and sufficient condition for the existence of such A_n, B_n in the case where Y has no normal component.     

Central limit theorem for stochastically continuous processes. Convergence to stable limit

LetX={X(t), t[0, 1]} be a stochastically continuous cadlag process. Assume that thek dimensional finite joint distributions ofX are in the domain of normal attraction of strictlyp-stable, 0<p<2, measure onR ^k for all 1k<. For functionsf, g such that _p (|X(x–X(u)|) >g(u–s) and _p (|X(s–X(t|)|X(t)–X(u|)>f(u–s), 0 s t u 1, conditions are found which imply that the distributions –(n ^–1/p (X ₁+···+X _n )),n1, converge weakly inD[0, 1] to the distribution of ap-stable process. HereX ₁,X ₂, ... are independent copies ofX and _p (Z)=sup _t<0 t ^pP{|Z|<t} denotes the weakpth moment of a random variable Z.     

A derivative-free optimization algorithm based on conditional moments

In this paper we propose a derivative-free optimization algorithm based on conditional moments for finding the maximizer of an objective function. The proposed algorithm does not require calculation or approximation of any order derivative of the objective function. The step size in iteration is determined adaptively according to the local geometrical feature of the objective function and a pre-specified quantity representing the desired precision. The theoretical properties including convergence of the method are presented. Numerical experiments comparing with the Newton, Quasi-Newton and trust region methods are given to illustrate the effectiveness of the algorithm.     

Operator-stable laws

Sharpe investigated the structure of full operator-stable measures μ on a vector group V and obtained decompositions, $\text{μ = μ}{}_1\text{1 μ}{}_2$ and V = V₁ ⊕ V₂, in terms of the Gaussian component μ₁ and the Poisson component μ₂. The subspaces V₁ and V₂ are here identified in terms of an exponent B for μ. Sharpe also pointed out that the Lévy measure M of μ is a mixture of Lévy measures concentrated on single orbits of t^B. Here, an explicit representation is obtained for M as such a mixture by constructing a measure on the unit sphere. Also, necessary and sufficient conditions are given that a Lévy measure be the Lévy measure of a full operator-stable measure. The final result deals with full Gaussian measures μ and establishes the connection between its covariance operator and the class of all exponents of μ.     

Conditional limit results for type I polar distributions

Let (S ₁,S ₂) = (R cos(Θ), R sin(Θ)) be a bivariate random vector with associated random radius R which has distribution function F being further independent of the random angle Θ. In this paper we investigate the asymptotic behaviour of the conditional survivor probability when u approaches the upper endpoint of F. On the density function of Θ we impose a certain local asymptotic behaviour at 0, whereas for F we require that it belongs to the Gumbel max-domain of attraction. The main result of this contribution is an asymptotic expansion of , which is then utilised to construct two estimators for the conditional distribution function . Furthermore, we allow Θ to depend on u.      

Conditional independence,conditional mixing and conditional association

Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.     

Operator geometric stable laws

Operator geometric stable laws are the weak limits of operator normed and centered geometric random sums of independent, identically distributed random vectors. They generalize operator stable laws and geometric stable laws. In this work we characterize operator geometric stable distributions, their divisibility and domains of attraction, and present their application to finance. Operator geometric stable laws are useful for modeling financial portfolios where the cumulative price change vectors are sums of a random number of small random shocks with heavy tails, and each component has a different tail index.     

A priori estimates for a singular limit approximation of the constitutive laws for chiral media in the time domain

The equations for the evolution of electromagnetic fields in chiral media, in the time domain, are nonlocal in time. In this work we study the validity of a singular limit (local in time) approximation for these nonlocal in time equations, by estimating the size of the difference of the fields as predicted by both models. In particular, we establish an a priori estimate for this difference, depending on the time horizon, properties of the domain, spatial properties of the initial data and the source terms and the chirality measure β of the approximating model.     

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.     

The existence of moments of the conditioned limit of the subcritical Crump-Mode-Jagers branching process

If Z (t) is the sum of the characteristics at time t of the population in a Crump-Mode-Jagers branching process, and T is the time to extinction, it is known that under certain conditions, the distribution of Z (t) conditioned on {T > t} converges to a proper distribution as t→∞. We derive necessary and sufficient conditions in terms of the offspring process, for the existence of integral moments of this limit distribution.     

Conditional versions of limit theorems for conditionally associated random variables

Relation between association and conditional association is answered, several examples show that the association of random variables does not imply the conditional association, and vice versa. Several fundamental properties of conditional associated random variables are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional Hájek-Rényi type inequalities, a conditional strong law of large numbers and a conditional central limit theorem stated in terms of conditional characteristic functions are established, which are conditional versions of the earlier results for associated random variables, respectively. In addition, some lemmas in the context are of independent interest.     

On conditional weak convergence

In this paper we discuss a number of technical issues associated with conditional weak convergence. The main modes of convergence of conditional probability distributions areuniform, probability, andalmost sure convergence in the conditioning variable. General results regarding conditional convergence are obtained, including details of sufficient conditions for each mode of convergence, and characterization theorems for uniform conditional convergence.     

On limit laws for sums of Pfeifer records

The problem of determining limiting distributions for sums of records has been studied by several authors who have considered a variety of assumptions sufficient to ensure that sums of records properly normalized will converge to a non-degenerate distribution. As a parallel to these endeavors, it is of interest to establish conditions under which the sum of Pfeifer records, properly normalized, converges. Pfeifer records are defined under the assumption that initial observations are i.i.d. with common survival function and following the (n−1)-th record value the observations are assumed to have survival function ,n=1,2,.... The study of the asymptotic behavior of sums of Pfeifer records constitutes a natural generalization of work on sums of classical records. The present paper introduces conditions under which the limit distribution of sums of Pfeifer records is non-degenerate.      

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface Sg of genus g grows asymptotically like

c(g)n5(g−1)/2−1γnn!     

Central limit theorem and stable laws for intermittent maps

In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form x+x¹⁺, for (0, 1). In particular, for >1/2, we show that the Birkhoff sums of a Hölder observable f converge to a normal law or a stable law, depending on whether f(0)=0 or f(0)0. The proof uses spectral techniques introduced by Sarig, and Wieners Lemma in non-commutative Banach algebras.Mathematics Subject Classification (2000):37A30, 37A50, 37C30, 37E05, 47A56, 60F05