In this paper we investigate the cluster behavior of linearly interacting Brownian motions indexed by . We show that (on a logarithmic scale) the block average process converges in path space to Brownian motion.

We construct a catalytic super process X (measure-valued spatial branching process) where the local branching rate is governed by an additive functional A of the motion process. These processes have been investigated before but under restrictive assumptions on A. Here we do not even need...

Classical super-Brownian motion (SBM) is known to take values in the space of absolutely continuous measures only if d=1. For d[greater-or-equal, slanted]2 its values are almost surely singular with respect to Lebesgue measure. This result has been generalized to more general motion laws and...

A strong law of large numbers is presented for a class of random variables X0, X1,..., which satisfy for a suitable function [latin small letter f with hook](x) > x.

We study asymptotic properties of non-negative random variables Xn, n[greater-or-equal, slanted]0, satisfying the recursion . If the functions g(x) and [sigma]2(x) are properly balanced at infinity, Xn is asymptotically [Gamma]-distributed in a suitable scale. This result contains several known...

We generalize a result by Kozlov on large deviations of branching processes (Zn) in an i.i.d. random environment. Under the assumption that the offspring distributions have geometrically bounded tails and mild regularity of the associated random walk S, the asymptotics of is (on logarithmic...

In this paper we investigate the problem of detecting a change in the drift parameters of a generalized Ornstein–Uhlenbeck process which is defined as the solution of <Equation ID="Equ23"> <EquationSource Format="TEX">$$\begin{aligned} dX_t=(L(t)-\alpha X_t) dt + \sigma dB_t \end{aligned}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink" display="block"> <mrow> <mtable columnspacing="0.5ex"> <mtr> <mtd columnalign="right"> <mrow> <mi>d</mi> <msub> <mi>X</mi> <mi>t</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi mathvariant="italic">α</mi> <msub> <mi>X</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>+</mo> <mi mathvariant="italic">σ</mi>...</mrow></mtd></mtr></mtable></mrow></math></equationsource></equationsource></equation>

We prove the complete convergence of (1/n)[Sigma]i < nh(Xi, Xn) for a square-integrable, degenerate kernel h. This is used to show that the Law of the Iterated Logarithm for degenerate U-statistics holds under a second moment condition.

It is well known that symmetric statistics based on a kernel with finite second moment have a limit law which can be described by a multiple Wiener-Ito integral. However, if the kernel has less than second moments, no weak limit law holds in general. In the present paper we show that by a...