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Hölder index for density states of (α,1,β)-superprocesses at a given point
A Hölder regularity index at given points for density states of (alpha,1,beta)-superprocesses with alpha>1+beta is determined. It is shown that this index is strictly greater than the optimal index of local Holder continuity for those density states.
Critical Galton-Watson processes: The maximum of total progenies within a large window
Consider a critical Galton-Watson process Z=Z_n: n=0,1,... of index 1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n in the window [k,...,k+j), and M_m(j) the maximum of the S_k with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EM_m(j) if the window width j=j_m satisfies that j/m converges in [0,1] as m tends to infinity. This will be achieved via establishing the asymptotic behavior of the tail probabilities of M_infinity(j).
Moderate deviations for random walk in random scenery
We investigate random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér's condition. We prove moderate deviation principles in dimensions d ≥ 2, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case d ≥ 4 we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In d ≥ 3, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst in $ = 2 we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.
On the left tail asymptotics for the limit law of supercritical Galton-Watson processes in the Böttcher case
Under a well-known scaling, supercritical Galton-Watson processes $Z$ converge to a non-degenerate non-negative random limit variable $W.$ We are dealing with the left tail (i.e. lose to the origin) asymptotics of its law. In the Bötcher case (i.e. if always at least two offspring are born), we describe the precise asymptotics exposing tiny oscillations (Theorem 1). Under a reasonable additional assumption, the oscillations disappear (Corollary 2). Also in the Böttcher case, we improve a recent lower deviation probability result by describing the precise asymptotics under a logarithmic scaling (Theorem 3). Under additional assumptions, we even get the fine (i.e. without log-scaling) asymptotics (Theorem 4).
Optimal Hölder index for density states of superprocesses with (1+[beta])-branching mechanism
For 0 < alpha leq 2, a super-alpha-stable motion X in R^d with branching of index 1 + beta in (1,2) is considered. If d < alpha / beta, a dichotomy for the density of states X_t at fixed times t > 0 holds: the density function is locally Hölder continuous if d = 1 and alpha > 1 + beta, but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal Hölder index.
Large deviations for sums defined on a Galton-Watson processes
In this paper we study the large deviation behavior of sums of i.i.d. random variables Xi defined on a supercritical Galton-Watson process Z. We assume the finiteness of the moments EX2 1 and EZ1 log Z1 . The underlying interplay of the partial sums of the Xi and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results on Z we recently published in [FW06].
Properties of states of super-[alpha]-stable motion with branching of index 1 + [beta]
It has been well-known for a long time that the measure states of the process in the title are absolutely continuous at any fixed time provided that the dimension of space is small enough. However, besides the very special case of one-dimensional continuous super-Brownian motion, properties of the related density functions were not well understood. Only in 2003, Mytnik and Perkins citeMytnikPerkins2003 revealed that in the Brownian motion case and if the branching is discontinuous, there is a dichotomy for the densities: Either there are continuous versions of them, or they are locally unbounded. We recently showed, that the same type of fixed time dichotomy holds also in the case of discontinuous motion. Moreover, the continuous versions are locally Hölder continuous, and we determined the optimal index for them. Finally, we determine the optimal index of Hölder continuity at given space points which is strictly larger than the optimal index of local Hölder continuity.