William E. Pruitt

Sample path properties of processes with stable components

SummaryIn this paper, processes in Rd of the form X(t)=(X1(t), X2(t), ⋯, XN(t), where Xi(t) is a stable process of index αi in Euclidean space of dimension di and d=d1 + ⋯ + dN, are considered. The asymptotic behaviour of the first passage time out of a sphere, and of the sojourn time in a sphere is established. Properties of the space-time process (X1(t), t) in Rd+1 are obtained when X1(t) is a stable process in Rd. For each of these processes, a Hausdorff measure function θ(h) is found such that the range set R(s) of the sample path on [0, s] has Hausdorff θ-measure c s for a suitable finite positive c.

Uniform Dimension Results for Processes with Independent Increments

Let X,(to) be a process in R a with stationary independent increments and let X(E, 09) denote the image under Xt(to) of a time set E. It is shown that dim X(E, og) 1 in R l is found to be a(1-1/~).

THE NUMBER OF EXTREME POINTS IN THE CONVEX HULL OF A RANDOM SAMPLE

Let (Xk} be an i.i.d. sequence taking values in R2 with the radial and spherical components independent and the radial component having a distribution with slowly varying tail. The number of extreme points in the convex hull of (XI, . *, X } is shown to have a limiting distribution which is obtained explicitly. Precise information about the mean and variance of the limit distribution is obtained.

Weak Convergence of Trimmed Sums

Considerable progress has been made in the last few years on problems concerning the asymptotic distribution of trimmed sums. The aim of this paper is to describe some of these results, discuss some of the problems which remain open, and provide a solution to one of these in an important special case.

An invariance principle for the local time of a recurrent random walk

SummaryLet (Sj) be a lattice random walk, i.e. Sj=X1 +...+Xj, where X1,X2,... are independent random variables with values in the integer lattice ℤ and common distribution F, and let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa% aaleaacaWGUbaabeaakiaacIcacqaHjpWDcaGGSaGaam4AaiaacMca% cqGH9aqpdaaeWbqaaiabeE8aJnaaBaaaleaacaGG7bGaam4Aaiaac2% haaeqaaaqaaiaadQgacqGH9aqpcaaIWaaabaGaamOBaiabgkHiTiaa% igdaa0GaeyyeIuoakiaacIcacaWGtbWaaSbaaSqaaiaadQgaaeqaaO% GaaiikaiabeM8a3jaacMcacaGGPaaaaa!508D!

An Integral Test for Subordinators

L_n (\omega ,k) = \sum\limits_{j = 0}^{n - 1} {\chi _{\{ k\} } } (S_j (\omega ))

Sample Functions of the

, the local time of the random walk at k before time n. Suppose EX1=0 and F is in the domain of attraction of a stable law G of index α> 1, i.e. there exists a sequence a(n) (necessarily of the form n1αl(n), where l is slowly varying) such that Sn/a(n)→ G. Define % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa% aaleaacaWGUbaabeaakiaacIcacqaHjpWDcaGGSaGaamyDaiaacMca% cqGH9aqpdaWcaaqaaiaadogacaGGOaGaamOBaiaacMcaaeaacaWGUb% aaaiaadYeadaWgaaWcbaGaamOBaaqabaGccaGGOaGaeqyYdCNaaiil% aiaacUfacaWG1bGaam4yaiaacIcacaWGUbGaaiykaiaac2facaGGPa% aaaa!4DCD!