We consider the power laws of certain limiting values in greedy lattice animals which were introduced by Cox, Gandolfi, Griffin, and Kesten (1993) and Gandolfi and Kesten (1994). We study the behavior of the limiting values as we change the parameter p.

Suppose that Xt = [summation operator][infinity]j=0cjZt-j is a stationary linear sequence with regularly varying cj's and with innovations {Zj} that have infinite variance. Such a sequence can exhibit both high variability and strong dependence. The quadratic form 89 plays an important role in...

It is shown that, if the initial measure is translation-invariant, then finite-range stochastic Ising models allowing zero flip-rates converge. In particular, the biased annihilating process converges to a mixture of a product measure and [delta]ø and the double-flipping process converges to a...

Markov properties of the solution to the wave equation in two spatial dimensions driven by a Lévy point process are considered. When the velocity of waves is 1, then for domains bounded by a plane, the sharp Markov property is shown to hold if and only if the angle between the plane and the...

We show the existence of a weakly self-attractive Brownian motion in dimensions two and three. In other words, we show the existence of a "polymer measure" that is formally defined by P(d[Omega]) = L-1 exp {[lambda][integral operator][integral operator]0 [less-than-or-equals, slant] s t...

We consider a process Yt which is the solution of a stochastic differential equation driven by a Lévy process with an initial condition Y0 = y0. We assume conditions under which Yt has a smooth density for any t 0. We consider a point y that the process can reach with a finite number of jumps...

From simple and natural assumptions, we study the functional asymptotic behavior in law of some random multilinear forms in martingale differences.

If B = (Bt)t [greater-or-equal, slanted] 0 is a standard Brownian motion started at x under Px for x [greater-or-equal, slanted] 0, and [tau] is any stopping time for B with Ex([tau]) < [infinity], then for each p > 1 the following inequality is shown to be sharp: The sharpness is realized through the stopping times of the...</[infinity],>

Let (Xij) be a double sequence of independent, identically distributed random variables, with mean zero and variance one, whose moment generating function is finite in a neighbourhood of the origin. Let Si(t) be the partial sum process constructed from Xi. We consider the sets Fn =...

Let X1, X2, ... be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that , if and only if E[X21] < [infinity] and E[X1] = 0. We prove that there are absolute constants C1, C2 [set membership, variant] (0, [infinity]) such that if X1, X2, ... are independent identically distributed mean zero random variables, then c1[lambda]-2 E[X12·1{X1[lambda]}][less-than-or-equals, slant]S([lambda])[less-than-or-equals, slant]C2[lambda]-2 E[X12·1{X1[lambda]}], for every [lambda] > 0.