Consider the radial projection onto the unit sphereof the path a $d$-dimensional Brownian motion $W$,started at the center of the sphere and run for unit time. Given the occupation measure $mu$ of this projectedpath, what can be said about the terminal point $W(1)$, or about therange of the...

A functional limit theorem is presented for the behaviour of Brownian motion conditioned to reach a high level during a fixed time interval. The asymptotic behaviour of the conditioned path as the level tends to infinity is related to Williams' path decomposition at the maximum.

Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the first n terms of the sequence, the next term equals the jth distinct value observed so far with probability pj,n, for j=1,2,... , and otherwise is a new value with...

Mark Kac introduced a method for calculating the distribution of the integral Av=[integral operator]0Tv(Xt) dt for a function v of a Markov process (Xt, t[greater-or-equal, slanted]0) and a suitable random time T, which yields the Feynman-Kac formula for the moment-generating function of Av. We...

The inverse first passage time problem asks whether, for a Brownian motion $B$ and a nonnegative random variable $\zeta$, there exists a time-varying barrier $b$ such that $\mathbb{P}\{B_sb(s),0\leq s\leq t\}=\mathbb{P}\{\zetat\}$. We study a "smoothed" version of this problem and ask whether...

We give an example of a central Wishart matrix W with one degree of freedom and scale matrix of rank 2 such that the diagonal entries of W are not associated. This allows us to conclude that no central Wishart matrix with one degree of freedom and scale matrix of rank greater than 1 is...

Filiz et al. (2008) proposed a model for the pattern of defaults seen among a group of firms at the end of a given time period. The ingredients in the model are a graph, where the vertices correspond to the firms and the edges describe the network of interdependencies between the firms, a...

We consider the elementary divisors and determinant of a uniformly distributed nxn random matrix with entries in the ring of integers of an arbitrary local field. We show that the sequence of elementary divisors is in a simple bijective correspondence with a Markov chain on the non-negative...