Raoul LePage

An iterated logarithm law for families of Brownian paths

SummaryIf n is large, a plot of log n independent Brownian paths over [0, n] is nearly certain to give the appearance of a shaded region having square root boundaries.

Addendum to: An iterated logarithm law for families of Brownian paths

After the papers [2] and [-3] had gone to press, the authors generalized these results to Brownian paths in separable Banach spaces and strengthened the limiting process to almost sure convergence. Through a communication with A. de Acosta we have learned that the latter result follows by an easy modification of the techniques used to prove Theorem 9 of [-1], which gives priority for this result to the authors of [,1]. Our proof (with V. Goodman) of the Banach space result is of some interest. The outer law, together with the one-dimensional inner law for the line passing through an arbitrary point on the surface of the unit ball of the kernel space, are used to deduce the full inner law by a simple compactness and scaling argument.

CONDITIONING VS. STANDARDIZATION FOR CONTRASTS ON ERRORS ATTRACTED TO STABLE LAWS

In multiple regression and other settings one encounters the problem of estimating sampling distributions for contrast operations applied to i.i.d. errors. Permutation bootstrap applied to least squares residuals has been proven to consistently estimate conditionalsampling distributions of contrasts, conditional upon order statistics of errors, even for long-tailed error distributions. How does this compare with the unconditional sampling distribution of the contrast when standardizing by the sample s.d. of the errors (or the residuals)? For errors belonging to the domain of attraction of a normal we present a limit theorem proving that these distributions are far closer to one another than they are to the limiting standard normal distribution. For errors attracted to α-stable laws with α ≤ 2 we construct random variables possessing these conditional and unconditional sampling distributions and develop a Poisson representation for their a.s. limit correlation ρα. We prove that ρ2= 1, ρα→ 1 for α → 0 + or 2 −, and ρα< 1 a.s. for α < 2.