Random Coefficient GARCH(1,1) Model With i.i.d. Coefficients

We introduce the GARCH(1,1) model with random i.i.d. coefficients. Conditions for the existence of a stationary solution of a random coefficient GARCH(1,1) equation are obtained. They generalize the well-known results of Nelson [14] and Terasvirta [18] in the case of constant (nonrandom) coefficients.__________Published in Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 467–480, October–December, 2004.     

Complete convergence and Cesàro summation for i.i.d. random variables

Summary Various results generalizing summation methods for divergent series of real numbers to analogous results for independent, identically distributed random variables have appeared during the last two decades. The main result of this paper provides necessary and sufficient conditions for the complete convergence of the Cesàro means of i.i.d random variables.     

Complete moment convergence for i.i.d. random variables

In the paper, the complete moment convergence is obtained for i.i.d. random variables such that all moments exist, but the moment generating function does not exist. The main results extend the related known works due to Gut and Stadtmüller.     

Laws of Large Numbers for Exchangeable Random Sets in Kuratowski-Mosco Sense

Abstract

Most of the results for laws of large numbers based on Banach space valued random sets assume that the sets are independent and identically distributed (IID) and compact, in which Rådström embedding or the refined method for collection of compact and convex subsets of a Banach space plays an important role. In this paper, exchangeability among random sets as a dependency, instead of IID, is assumed in obtaining strong laws of large numbers, since some kind of dependency of random variables may be often required for many statistical analyses. Also, the Hausdorff convergence usually used is replaced by another topology, Kuratowski-Mosco convergence. Thus, we prove strong laws of large numbers for exchangeable random sets in Kuratowski-Mosco convergence, without assuming the sets are compact, which is weaker than Hausdorff sense.     

Trimmed sums of i.i.d. Banach space valued random variables

Limit laws for trimmed sums of triangular arrays of i.i.d. Banach space valued random variables are studied. It is shown that if the array belongs to the domain of attraction of an infinitely divisible law without Gaussian component on a separable Banach space of type 2, then the trimmed sum converges weakly to a nondegenerate Banach space valued random variable.     

关于独立同分布正态随机变量部分和尾概率的注(英文)

设{X,Xn,n≥1}是独立同分布正态随机变量序列,EX=0且EX2=σ2>0,Sn=sum (Xk) form k=1 to n,λ(ε) =sum form (P(|Sn|≥ nε)) form n=1 to ∞.在本文中,我们证明了存在正常数C1和C2,使得对足够小的ε>0,成立下列不等式C1ε3 ≤ε2λ(ε)-σ2+ε2 /2 ≤ C2ε3.     

Tightness of products of i.i.d. random matrices

Summary In this paper we present a necessary and sufficient condition for tightness of products of i.i.d. finite dimensional random nonnegative matrices. We give an example illustrating the use of our theorem and treat completely the case of 2×2 matrices. We also describe stationary solutions of the linear equationy _n=X_ny_n–1, n>0, in (R ^d )⁺, whereX ₁,X ₂,... are i.i.d.d×d nonnegative matrices.     

Strong approximations for partial sums of i.i.d.B-valued r.v.'s in the domain of attraction of a Gaussian law

Summary We obtain a strong approximation theorem for partial sums of i.i.d.d-dimensional r.v.'s with possibly infinite second moments. Using this result, we can extend Philipp's strong invariance principle for partial sums of i.i.d.B-valued r.v.'s satisfying the central limit theorem toB-valued r.v.'s which are only in the domain of attraction of a Gaussian law. This new strong invariance principle implies a compact as well as a functional law of the iterated logarithm which improve some recent results of Kuelbs (1985).     

An Inequality for Large Deviation Probabilities of Sums of Bounded i.i.d. Random Variables

We provide precise bounds for tail probabilities, say {M _n x}, of sums M _n of bounded i.i.d. random variables. The bounds are expressed through tail probabilities of sums of i.i.d. Bernoulli random variables. In other words, we show that the tails are sub-Bernoullian. Sub-Bernoullian tails are dominated by Gaussian tails. Possible extensions of the methods are discussed.     

A remark on LSL for weighted sums of i.i.d random elements

In the paper, the upper bound and lower bound of the law of the single logarithm (LSL) are established under the condition that the sequence of the normalized weighted sums of random elements is bounded in probability. The main result improves the upper bound in [Sung, S.H., 2009. A law of the single logarithm for weighted sums of i.i.d. random elements. Statist. Probab. Lett., 79, 1351–1357] and hence extends the result in [Chen, P., Gan, S., 2007. Limiting behavior of weighted sums of i.i.d. random variables. Statist. Probab. Lett., 77, 1589–1599].     

Upper packing dimension of a measure and the limit distribution of products of i.i.d. stochastic matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. 2 × 2 stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is countably infinite. It extends a previous result for which the support of the random matrices is finite. The result is based on adapting existing proofs in the context of attractors and iterated function systems to the case of infinite iterated function systems.     

A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment

We consider a random walk on $\mathbb{Z }^d,\ d\ge 2$ , in an i.i.d. balanced random environment, that is a random walk for which the probability to jump from $x\in \mathbb{Z }^d$ to nearest neighbor $x+e$ is the same as to nearest neighbor $x-e$ . Assuming that the environment is genuinely $d$ -dimensional and balanced we show a quenched invariance principle: for $P$ almost every environment, the diffusively rescaled random walk converges to a Brownian motion with deterministic non-degenerate diffusion matrix. Within the i.i.d. setting, our result extend both Lawler’s uniformly elliptic result (Comm Math Phys, 87(1), pp 81–87, 1982/1983) and Guo and Zeitouni’s elliptic result (to appear in PTRF, 2010) to the general (non elliptic) case. Our proof is based on analytic methods and percolation arguments.     

Almost sure rates of mixing for i.i.d. unimodal maps

It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and Keller-Nowicki proved exponential decay of its correlation functions. Benedicks and Young [8], and Baladi and Viana [4] studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical properties of the sample stationary measures of i.i.d. itineraries are more difficult to estimate than the “averaged statistics”. Adapting to random systems, on the one hand partitions associated to hyperbolic times due to Alves [1], and on the other a probabilistic coupling method introduced by Young [26] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing.     

A note on occurrence of gapped patterns in i.i.d. sequences

A new martingale technique is developed to find formulas for the expected value and generating function of the waiting time until one observes a gapped pattern (or a structured motif) in an i.i.d. sequence of random letters from a finite alphabet.     

FRACTAL PROPERTIES OF POLAR SETS OF RANDOM STRING PROCESSES

This paper studies fractal properties of polar sets for random string processes. We give upper and lower bounds of the hitting probabilities on compact sets and prove some sufficient conditions and necessary conditions for compact sets to be polar for the random string process. Moreover, we also determine the smallest Hausdorff dimensions of non-polar sets by constructing a Cantor-type set to connect its Hausdorff dimension and capacity.     

Infinite Divisibility of Random Objects in Locally Compact Positive Convex Cones

Random objects taking on values in a locally compact second countable convex cone are studied. The convex cone is assumed to have the property that the class of continuous additive positively homogeneous functionals is separating, an assumption which turns out to imply that the cone is positive. Infinite divisibility is characterized in terms of an analog to the Lévy–Khinchin representation for a generalized Laplace transform. The result generalizes the classical Lévy–Khinchin representation for non-negative random variables and the corresponding result for random compact convex sets inRⁿ. It also gives a characterization of infinite divisibility for random upper semicontinuous functions, in particular for random distribution functions with compact support and, finally, a similar characterization for random processes on a compact Polish space.     

ON THE EXPECTATION AND VARIANCE OF HAMMING DISTANCE BETWEEN TWO I.I.D RANDOM VECTORS

1.IntroductionLetFZn={0,1}"beann-dimensionalvectorspaceoverthebinaryfieldFZ={0,1}.TheHammingdistancebetweentwovectorsx=(xl,'tx.)andy=(yi,'ty.)isthenumberofcoordinateswheretheydiffer,andisdenotedbydH(x,y),dH(X,y)=ZIXi~ail.i=1TheHammingweightofxisthenumberofnon-zerocoordinates,andisdenotedbyWH(x).ObviouslyAH(x)=dH(x,0),where0isthezerovector.Thescalarproductofxandyis(x,y)=xlyl ' xestinF2.ForasetAgFZn,IAIdenotesthecardinalityofA.TheaveragedistanceinAisdefinedby*Thisresearchissupp…     

The k-record processes are i.i.d.

Let X ₁, X ₂, ... be i.i.d. positive random variables, and let _n be the initial rank of X _n (that is, the rank of X _n among X ₁, ..., X _n). Those observations whose initial rank is k are collected into a point process N ^k on ⁺, called the k-record process. The fact that {itN^k; k=1, 2, ... are independent and identically distributed point processes is the main result of the paper. The proof, based on martingales, is very rapid. We also show that given N ¹, ..., N ^k, the lifetimes in rank k of all observations of initial rank at most k are independent geometric random variables.These results are generalised to continuous time, where the analogue of the i.i.d. sequence is a time-space Poisson process. Initially, we think of this Poisson process as having values in ⁺, but subsequently we extend to Poisson processes with values in more general Polish spaces (for example, Brownian excursion space) where ranking is performed using real-valued attributes.     

Global random attractors are uniquely determined by attracting deterministic compact sets

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic. Entrata in Redazione il 10 marzo 1997.     

Multivariate Hazard Rates under Random Censorship

The data consists of multivariate failure times under right random censorship. By the kernel smoothing technique, convolutions of cumulative multivariate hazard functions suggest estimators of the so-called multivariate hazard functions. We establish strong i.i.d. representations and uniform bounds of the remainder terms on some compact sets of the underlying space. Thus asymptotic normality and uniform consistency on such sets are obtained. The asymptotic mean squared error gives an optimal bandwidth by the plug-in method. Simulations assess the performance of our estimators.