Philip S. Griffin

The most visited site of Brownian motion and simple random walk

SummaryLet L(t, x) be the local time at x for Brownian motion and for each t, let

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

\bar V(t) = \inf \{ x\underline{\underline > } 0;L(t,x) \vee L(t, - x) = \mathop {\sup }\limits_y L(t,y)\}

Path decomposition of ruinous behavior for a general Lévy insurance risk process.

, the absolute value of the most visited site for Brownian motion up to time t. In this paper we prove that ¯V(t) is transient and obtain upper and lower bounds for the rate of growth of ¯V(t). The main tools used are the Ray-Knight theorems and Williams path decomposition of a diffusion. An invariance principle is used to get analogous results for simple random walks. We also obtain a law of the iterated logarithm for ¯V(t).

The influence of extremes on the law of the iterated logarithm

Let X 1 , …, X n be a sequence of non-degenerate, symmetric, independent identically distributed random variables, and let S n ( r n ) denote their sum when the r n largest in modulus have been removed. We obtain necessary and sufficient conditions for asymptotic normality of the studentized version of S n ( r n ), and compare this to the condition for asymptotic normality of the scalar normalized version. In particular, when r n = r these conditions are the same, but when r n → ∞the former holds more generally.

Weak Convergence of Trimmed Sums

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.

Small and large time stability of the time taken for a Lévy process to cross curved boundaries

SummaryLet X(t) be a symmetric stable process of index α> 1 with local time L(t, x) and define R(t)=¦{x:X(s)=x for some s≦t}¦ and L*(t) = sup{L(t, x): x∈ℝ1}. We prove that

Asymptotic normality of winsorized means

\begin{gathered} \mathop {\lim \sup }\limits_{t \to \infty } t^{ - 1/\alpha } (\log \log {\text{t}})^{ - (1 - 1/\alpha )} R(t) = c_1 {\text{ a}}{\text{.s}}{\text{.}} \hfill \\ \mathop {\lim \inf }\limits_{t \to \infty } t^{ - (1 - 1/\alpha )} (\log \log {\text{t}})^{1 - 1/\alpha } L*(t) = c_2 {\text{ a}}{\text{.s}}{\text{.}} \hfill \\ \end{gathered}