In this paper, we study a version of the law of the logarithm in a Banach space setting. Some necessary and some sufficient conditions are presented for the law of the logarithm for B-valued random variables. The law of the logarithm, the law of the iterated logarithm and the central limit...

We consider some deterministic cellular automata on the state space {0, 1}d, starting from the product of Bernoulli measures and evolving in discrete time according to the bootstrap percolation rules, in which a 0 changes to a 1 when it has at least l neighbours which are 1. We prove that in...

Suppose that (X(n)) is a finite adapted sequence of d-dimensional random variables defined on some filtered probability space ([Omega], F, (Fn), P). We obtain conditions which are necessary and sufficient for the existence of a probability measure Q equivalent to P (which we call an equivalent...

Some probability inequalities are obtained, and some liminf results are established for a two-parameter Wiener process by using these inequalities. The results obtained improve those of Lacey (1989) and get the watershed between the Chung type laws of the iterated logarithm and the Lacey type...

For the nearest-neighbor Potts ferromagnets on 2, it is shown that the magnetized states are characterized by percolation of the preferred species, which generalizes a result known previously only for the Ising system.

We characterize the upper and lower functions of a real-valued Wiener process normalized by the supremum of its local times.

Let M = {Mz, z [epsilon] R2+} be a two-parameter strong martingale, A be a two-parameter increasing process on R2+ = [0, + [infinity]) x [0, + [infinity]). Consider the following stochastic differential equations in the plane: for z [epsilon] R2+. Under some assumptions on the coefficients a, b...

Let {Xn, n = 0} be a Markov chains with the state space S = {1, 2, ..., m}, and the probability distribution P(x0) [Pi]nk=1Pk(xkxk-1), where Pk(ji) is the transition probability P(Xk = jXk-1 = i). Let gk(i, j) be the functions defined on S x S, and let Fn([omega]) = (1/n)[Sigma]nk=1gk(Xk-1, Xk)....

We study financial market incompleteness induced by discontinuities in asset returns. When there are multiple outcomes for a discontinuity, it is shown that this incompleteness cannot be removed by the introduction of extra securities. Claims cannot be hedged and are thereby not uniquely priced...

Let [beta] be a positive number: we consider a particle performing a one-dimensional Brownian motion with drift -[beta], diffusion coefficient 1, and a reflecting barrier at 0. We prove that the time R, needed by the particle to reach a random level X, has the same distribution tails as...