Sandor Csorgo

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

SummaryConsider the stationary sequenceX1=G(Z1),X2=G(Z2),..., whereG(·) is an arbitrary Borel function andZ1,Z2,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z1Zk+1) satisfyingr(0)=1 and ∑k=1∞ |r(k)|m < ∞, where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {Xn}, the integerm is the Hermite rank of the family {I{G(·)≦ x} −F(x):x∈R}. LetFn(·) be the empirical distribution function ofX1,...,Xn. We prove that, asn→∞, the empirical processn1/2{Fn(·)-F(·)} converges in distribution to a Gaussian process in the spaceD[−∞,∞].

Random-design regression under long-range dependent errors

We consider the random-design nonparametric regression model with long-range dependent errors that may also depend on the independent and identically distributed explanatory variables. Disclosing a smoothing dichotomy, we show that the finite-dimensional distributions of the Nadaraya-Watson kernel estimator of the regression fimunction converge either to those of a degenerate process with completely dependent marginals or to those of a Gaussian white-noise process. The first case occurs when the bandwidths are large enough in a specified sense to allow long-range dependence to prevail. The second case is for bandwidths that are small in the given sense, when both the required norming sequence and the limiting process are the same as if the errors were independent. This conclusion is also derived for all bandwidths if the errors are short-range dependent. The borderline situation results in a limiting convolution of the two cases. The main results contrast with previous findings for deterministic-design regression.

A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games

Extending a result of Einmahl, Haeusler and Mason (1988), a characterization of the almost sure asymptotic stability of lightly trimmed sums of upper order statistics is given when the right tail of the underlying distribution with positive support is surrounded by tails that are regularly varying with the same index. The result is motivated by applications to cumulative gains in a sequence of generalized St. Petersburg games in which a fixed number of the largest gains of the player may be withheld.

On the law of large numbers for the bootstrap mean

Direct and elementary proofs are given for weak and strong laws of large numbers for bootstrap sample means under minimal moment conditions. Concerning the required rate of divergence of the bootstrap sample size, the strong laws obtained improve on those of Athreya (1983). Ams 1980 Subject Classifications: Primary 60F15; Secondary 62G05

Distant long-range dependent sums and regression estimation

Consider a stationary sequence G(Z0), G(Z1), ..., where G(·) is a Borel function and Z0, Z1, ... is a sequence of standard normal variables with covariance function E(Z0Zj) = j-[alpha]L(j), j = 1, 2, ..., where E(G(Z0)) = 0, E(G2(Z0)) 0 and sequences of gap-lengths l1,n, ..., lk,n such that l1,n --> [infinity] and lj,n - lj-1,n --> [infinity], j = 2, ..., k, arbitrary slowly, the vector process , 0

Density estimation in the simple proportional hazards model

In the simple proportional hazards model of random right censorship the limiting variance vACL2(x) at x of the kernel density estimator based on the Abdushukurov-Cheng-Lin estimator is shown to be equal to the corresponding variance pertaining to the Kaplan-Meier estimator times the expected proportion p of uncensored observations. More surprisingly, for appropriate p, vACL2(x) is smaller than the asymptotic variance of the classical kernel estimator based on a complete sample, for any x below the (1 - e-1)-quantile.

Random Graphs and the Strong Convergence of Bootstrap Means

We consider graphs G n generated by multisets [Iscr ] n with n random integers as elements, such that vertices of G n are connected by edges if the elements of [Iscr ] n that the vertices represent are the same, and prove asymptotic results on the sparsity of edges connecting the different subgraphs G n of the random graph generated by ∪ ∞ n =1 [Iscr ] n . These results are of independent interest and, for two models of the bootstrap, we also use them here to link almost sure and complete convergence of the corresponding bootstrap means and averages of related randomly chosen subsequences of a sequence of independent and identically distributed random variables with a finite mean. Complete convergence of these means and averages is then characterized in terms of a relationship between a moment condition on the bootstrapped sequence and the bootstrap sample size. While we also obtain new sufficient conditions for the almost sure convergence of bootstrap means, the approach taken here yields the first necessary conditions.

Intermediate- and extreme-sum processes

Let X1,n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn,n be the order statistics of n independent random variables with a common distribution function F and let kn be positive numbers such that kn --> [infinity] and . With suitable centering and norming, we investigate the weak convergence of the intermediate-sum process [summation operator]i=[left ceiling]akn[right ceiling]+1[left ceiling]tkn[right ceiling]Xn+1-i,n, a [less-than-or-equals, slant] t [less-than-or-equals, slant] b, where 0

Intermediate sums and stochastic compactness of maxima

Abstract A connection between the convergence in distribution of sums of intermediate order statistics and the stochastic compactness of maxima is established.

On Sums of Overlapping Products of Independent Bernoulli Random Variables

We find the exact distribution of an arbitrary remainder of an infinite sum of overlapping products of a sequence of independent Bernoulli random variables.