Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.     

Asymptotics of Dominated Gaussian Maxima

Let {X _n , n1} be a sequence of independent Gaussian random vectors in R ^d d2. In this paper an asymptotic evaluation of P{max₁in X _i a _n Z+b _n } with Z another Gaussian random vector is obtained for a _n, b _n R ^d two vectors obeying certain conditions.     

On characterization of multivariate stable distributions via random linear statistics

LetX ₁,X ₂, ...,X _n be independent and identically distributed random vectors inR ^d , and letY=(Y ₁,Y ₂, ...,Y _n )′ be a random coefficient vector inR ⁿ , independent ofX _j ^/′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.     

Convergence to type I distribution of the extremes of sequences defined by random difference equation

We study the extremes of a sequence of random variables (R_n) defined by the recurrence R_n=M_nR_n−1+q, n≥1, where R₀ is arbitrary, (M_n) are iid copies of a non-degenerate random variable M, 0≤M≤1, and q>0 is a constant. We show that under mild and natural conditions on M the suitably normalized extremes of (R_n) converge in distribution to a double-exponential random variable. This partially complements a result of de Haan, Resnick, Rootzén, and de Vries who considered extremes of the sequence (R_n) under the assumption that P(M>1)>0.     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

Sharp estimates for the CDF of quadratic forms of MPE random vectors

In this paper we develop an efficient analytical expansion of the cumulative distribution function (cdf) XBX^t where X=(X₁,…,X_n+1) with n≥2, follows a multivariate power exponential distribution (MPE). Our approach provides a sharp estimate of the cumulative distribution function of a quadratic form of MPE, together with explicit error estimates.     

General theorems on rates of convergence in distribution of random variables I. General limit theorems

Let (X_n)_n?$\text{N}$ be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions F_Xn, and S_n = Σ_i=1ⁿX_i. The authors present limit theorems together with convergence rates for the normalized sums ?(n)S_n, where ?: $\text{N}$ → $\text{R}$⁺, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝_$\text{R}$f(x) d[F_?(n)Sn(x) ? F_X(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.     

Conditional ordering of order statistics

For any positive integers m and n, let X₁,X₂,…,X_m∨n be independent random variables with possibly nonidentical distributions. Let X_1:n≤X_2:n≤?≤X_n:n be order statistics of random variables X₁,X₂,…,X_n, and let X_1:m≤X_2:m≤?≤X_m:m be order statistics of random variables X₁,X₂,…,X_m. It is shown that (X_j:n,X_j+1:n,…,X_n:n) given X_i:m>y for j−i≥max{n−m,0}, and (X_1:n,X_2:n,…,X_j:n) given X_i:m≤y for j−i≤min{n−m,0} are all increasing in y with respect to the usual multivariate stochastic order. We thus extend the main results in Dubhashi and Häggström (2008) [1] and Hu and Chen (2008) [2].     

Asymptotics of random contractions

In this paper we discuss the asymptotic behaviour of random contractions X=RS, where R, with distribution function F, is a positive random variable independent of S∈(0,1). Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.     

Deformations of the generalised Picard bundle

Let X be a nonsingular algebraic curve of genus g?3, and let M_ξ denote the moduli space of stable vector bundles of rank n?2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g=3 then n?4 and if g=4 then n?3, and suppose further that n₀, d₀ are integers such that n₀?1 and nd₀+n₀d>nn₀(2g-2). Let E be a semistable vector bundle over X of rank n₀ and degree d₀. The generalised Picard bundle W_ξ(E) is by definition the vector bundle over M_ξ defined by the direct image where U_ξ is a universal vector bundle over X×M_ξ. We obtain an inversion formula allowing us to recover E from W_ξ(E) and show that the space of infinitesimal deformations of W_ξ(E) is isomorphic to H¹(X,End(E)). This construction gives a locally complete family of vector bundles over M_ξ parametrised by the moduli space M(n₀,d₀) of stable bundles of rank n₀ and degree d₀ over X. If (n₀,d₀)=1 and W_ξ(E) is stable for all E∈M(n₀,d₀), the construction determines an isomorphism from M(n₀,d₀) to a connected component M⁰ of a moduli space of stable sheaves over M_ξ. This applies in particular when n₀=1, in which case M⁰ is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.     

A universal strong law of large numbers for conditional expectations via nearest neighbors

For k_n-nearest neighbor estimates of a regression Y on X (d-dimensional random vector X, integrable real random variable Y) based on observed independent copies of (X,Y), strong universal pointwise consistency is shown, i.e., strong consistency P_X-almost everywhere for general distribution of (X,Y). With tie-breaking by indices, this means validity of a universal strong law of large numbers for conditional expectations E(Y|X=x).     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Empirical processes of multidimensional systems with multiple mixing properties

We establish a multivariate empirical process central limit theorem for stationary R^d-valued stochastic processes (X_i)_i≥1 under very weak conditions concerning the dependence structure of the process. As an application, we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling et al. (2009) [9] in the univariate case. As an important technical ingredient, we prove a 2pth moment bound for partial sums in multiple mixing systems.     

Recurrent random walks in nonnegative matrices II

Summary In this paper, we continue the study undertaken in our earlier paper [M1]. One of the main results here can be described as follows. LetX ₀,X ₁, ... be a sequence of iid random affine maps from (R ⁺) ^d into itself. Let us write:W _n X _n X _n –1...X ₀ andZ _n X ₀ X ₁...X _n , where composition of maps is the rule of multiplication. By the attractorA(u),u(R ⁺) ^d , we mean the setA u={y(R⁺)^d:P(W_n uN i.o.) > 0 for every openN containingy}. It is shown that the attractorA(u), under mild conditions, is the support of a stationary probability measure, when the random walk (Z _n ) has at least one recurrent state.     

A central limit theorem for generalized multilinear forms

Let X₁, …, X_n be independent random variables and define for each finite subset I {1, …, n} the σ-algebra = σ{X_i : i ε I}. In this paper -measurable random variables W_I are considered, subject to the centering condition E(W_I ) = 0 a.s. unless I J. A central limit theorem is proven for d-homogeneous sums W(n) = Σ_{I = d}W_I, with var W(n) = 1, where the summation extends over all (_n^d) subsets I {1, …, n} of size I = d, under the condition that the normed fourth moment of W(n) tends to 3. Under some extra conditions the condition is also necessary.     

On the observation closest to the origin

Out of n i.i.d. random vectors in $\text{R}$^d let X¹_n be the one closest to the origin. We show that X¹_n has a nondegenerate limit distribution if and only if the common probability distribution satisfies a condition of multidimensional regular variation. The result is then applied to a problem of density estimation.     

Reduction of variance for Gaussian densities via restriction to convex sets

Let X be a random vector with values in $\text{R}$ⁿ and a Gaussian density f. Let Y be a random vector whose density can be factored as k · f, where k is a logarithmically concave function on $\text{R}$ⁿ. We prove that the covariance matrix of X dominates the covariance matrix of Y by a positive semidefinite matrix. When k is the indicator function of a compact convex set A of positive measure the difference is positive definite. If A and X are both symmetric Var(a · X) is bounded above by an expression which is always strictly less than Var(a · X) for every a ∈ $\text{R}$ⁿ. Finally some counterexamples are given to show that these results cannot be extended to the general case where f is any logarithmically concave density.     

Exponential families of mixed Poisson distributions

If I=(I₁,…,I_d) is a random variable on [0,∞)^d with distribution μ(dλ₁,…,dλ_d), the mixed Poisson distribution MP(μ) on N^d is the distribution of (N₁(I₁),…,N_d(I_d)) where N₁,…,N_d are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)^d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v₀+v₁Y₁+?+v_qY_q for some q?d, for some vectors v₀,…,v_q of [0,∞)^d with disjoint supports and for independent standard real gamma random variables Y₁,…,Y_q.