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)) < [infinity], 0 < [alpha] < 1 and L(·) varies slowly at infinity. Let Sn(t) = [summation operator][lower left corner]nt[right floor]-1j=0 G(Zj), t [greater-or-equal, slanted] 0, be the associated partial-sum process. The main result is that for any fixed and 0 < b < [infinity], a suitable norming sequence an > 0 and sequences of gap-lengths l1,n, ..., lk,n such that...</[infinity],>

We show that the accumulated gain Sn and the maximal gain Mn in n St. Petersburg games satisfy almost sure limit theorems with nondegenerate limits, even though ordinary asymptotic distributions do not exist for Sn and Mn with any numerical centering and norming sequences.

A nonparametric large sample test is proposed for testing the linearity of a regression model with independent and identically distributed errors satisfying only a very mild tail condition. The statistic is based on the functional least squares estimator of the slope vector. The test is applied...

An intuitively appealing exact formula is derived for the expected total winnings in n generalized St. Petersburg games when the bank withholds the m largest winnings.

Tests of total independence of d (>=2) random variables are proposed using the empirical characteristic function. The approach is parallel to that of Hoeffding, Blum, Kiefer, and Rosenblatt.

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...

Almost sure conditional complete convergence of bootstrap means is considered with inside and outside rates of the type of Spitzer and of Baum and Katz. As complete analogues of Gut and Spataru's theorems for ordinary means, precise asymptotic behavior is determined for bootstrap sample sizes...

Multivariate integral kernel transformations of the multivariate empirical process are considered. The asymptotic behaviour of these transforms are investigated when a null-hypothesis completely specifies the underlying distribution and also when parameters are also estimated from the sample. In...

Shorack [9] gave more intelligible equivalent form of O'Reilly's [4] necessary and sufficient condition for the weak convergence of the uniform quantile process in weighted sup-norm metrics, and he derived the sufficiency part of O'Reilly's theorem from a strong approximation result of M....

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...