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1.
Let X
i
, i∈N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a mapping B→R. Under a central limit theorem assumption, an asymptotic evaluation of Z
n
= E (exp (n
Φ (∑
i
=1
n
X
i
/n))), up to a factor (1 + o(1)), has been gotten in Bolthausen [1]. In this paper, we show that the same asymptotic evaluation can be gotten without
the central limit theorem assumption.
Received: 19 September 1997 / Revised version:22 April 1999 相似文献
2.
Jérôme Dedecker 《Probability Theory and Related Fields》1998,110(3):397-426
Summary. We prove a central limit theorem for strictly stationary random fields under a projective assumption. Our criterion is similar
to projective criteria for stationary sequences derived from Gordin's theorem about approximating martingales. However our
approach is completely different, for we establish our result by adapting Lindeberg's method. The criterion that it provides
is weaker than martingale-type conditions, and moreover we obtain as a straightforward consequence, central limit theorems
for α-mixing or φ-mixing random fields.
Received: 19 February 1997 / In revised form: 2 September 1997 相似文献
3.
Neil O'Connell 《Probability Theory and Related Fields》1998,110(3):277-285
Summary. We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with
small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique.
The proof yields an asymptotic formula for the probability that the random graph is connected.
We also present an LDP and related result for the number of isolated vertices. Here we make use of a simple but apparently
unknown characterisation, which is obtained by embedding the random graph in a random directed graph. The results demonstrate
that, at this scaling, the properties `connected' and `contains no isolated vertices' are not asymptotically equivalent. (At
the threshold probability they are asymptotically equivalent.)
Received: 14 November 1996 / In revised form: 15 August 1997 相似文献
4.
Agoston Pisztora Tobias Povel Ofer Zeitouni 《Probability Theory and Related Fields》1999,113(2):191-219
ωx } (taking values in the interval [1/2, 1)), which serve as an environment. This environment defines a random walk {X k } (called a RWRE) which, when at x, moves one step to the right with probability ω x , and one step to the left with probability 1 −ωx. Solomon (1975) determined the almost-sure asymptotic speed (= rate of escape) of a RWRE, in a more general set-up. Dembo, Peres and Zeitouni (1996), following earlier work by Greven and den Hollander (1994) on the quenched case, have computed rough tail asymptotics for the empirical mean of the annealed RWRE. They conjectured the form of the rate function in a full LDP. We prove in this paper their conjecture. The proof is based on a “coarse graining scheme” together with comparison techniques. Received: 22 July 1997/Revised version: 15 June 1998 相似文献
5.
Alexander D. Wentzell 《Probability Theory and Related Fields》1999,113(2):255-271
. For a certain class of families of stochastic processes ηε(t), 0≤t≤T, constructed starting from sums of independent random variables, limit theorems for expectations of functionals F(ηε[0,T]) are proved of the form
where w
0 is a Wiener process starting from 0, with variance σ2 per unit time, A
i
are linear differential operators acting on functionals, and m=1 or 2. Some intricate differentiability conditions are imposed on the functional.
Received: 12 September 1995 / Revised version: 6 April 1998 相似文献
6.
Hans-Peter Scheffler 《Probability Theory and Related Fields》2000,116(2):257-271
For a random vector belonging to the (generalized) domain of operator semistable attraction of some nonnormal law we prove
various variants of Chover's law of the iterated logarithm for the partial sum. Furthermore we also derive some large deviation
results necessary for the proof of our main theorems.
Received: 30 September 1998 / Revised version: 28 May 1999 相似文献
7.
Thomas M. Liggett 《Probability Theory and Related Fields》1996,106(4):495-519
Summary. Branching random walks and contact processes on the homogeneous tree in which each site has d+1 neighbors have three possible types of behavior (for d≧ 2): local survival, local extinction with global survival, and global extinction. For branching random walks, we show that
if there is local extinction, then the probability that an individual ever has a descendent at a site n units away from that individual’s location is at most d
− n/2
, while if there is global extinction, this probability is at most d
−n
. Next, we consider the structure of the set of invariant measures with finite intensity for the system, and see how this
structure depends on whether or not there is local and/or global survival. These results suggest some problems and conjectures for contact processes on trees. We prove some and
leave others open. In particular, we prove that for some values of the infection parameter λ, there are nontrivial invariant measures which have a density tending to zero in all directions, and hence are different
from those constructed by Durrett and Schinazi in a recent paper.
Received: 26 April 1996/In revised form: 20 June 1996 相似文献
8.
R. A. Doney 《Probability Theory and Related Fields》1997,107(4):451-465
Summary. If {S
n
,n≧0} is an integer-valued random walk such that S
n
/a
n
converges in distribution to a stable law of index α∈ (0,1) as n→∞, then Gnedenko’s local limit theorem provides a useful estimate for P{S
n
=r} for values of r such that r/a
n
is bounded. The main point of this paper is to show that, under certain circumstances, there is another estimate which is
valid when r/a
n
→ +∞, in other words to establish a large deviation local limit theorem. We also give an asymptotic bound for P{S
n
=r} which is valid under weaker assumptions. This last result is then used in establishing some local versions of generalized
renewal theorems.
Received: 9 August 1995 / In revised form: 29 September 1996 相似文献
9.
David Steinsaltz 《Probability Theory and Related Fields》1997,107(1):99-121
Summary. A self-modifying random walk on is derived from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed. This
process converges almost surely to a random variable which is totally singular with respect to Lebesgue measure, and which
is supported on a subset of having Hausdorff dimension less than , which we calculate by a theorem of Billingsley. By generating function techniques we then calculate the exponential rate
of convergence of the process to its limit point, which may be taken as a bound for the convergence of the measure in the
Wasserstein metric. We describe how the process may viewed as a random walk on the space of monotone piecewise linear functions,
where moves are taken by successive compositions with a randomly chosen such function.
Received: 20 November 1995 / In revised form: 14 May 1996 相似文献
10.
We present an upper bound O(n
2
) for the mixing time of a simple random walk on upper triangular matrices. We show that this bound is sharp up to a constant,
and find tight bounds on the eigenvalue gap. We conclude by applying our results to indicate that the asymmetric exclusion
process on a circle indeed mixes more rapidly than the corresponding symmetric process.
Received: 25 January 1999 / Revised version: 17 September 1999 / Published online: 14 June 2000 相似文献
11.
Ito's rule is established for the diffusion processes on the graphs. We also consider a family of diffusions processes with
small noise on a graph. Large deviation principle is proved for these diffusion processes and their local times at the vertices.
Received: 12 February 1997 / Revised version: 3 March 1999 相似文献
12.
John Hawkes 《Probability Theory and Related Fields》1998,112(1):1-11
Exact results are proved for the capacity of pullbacks of analytic sets by stable processes.
Received: 25 May 1988 / Revised version: 15 September 1997 相似文献
13.
14.
Suppose K is a compact convex set in ℝ2 and X
i
, 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X
i
we study the asymptotic properties of certain statistics of the convex hull of the sample.
Received: 24 July 1996/Revised version: 24 February 1998 相似文献
15.
16.
We prove that the process of the most visited site of Sinai's simple random walk in random environment is transient. The
rate of escape is characterized via an integral criterion. Our method also applies to a class of recurrent diffusion processes
with random potentials. It is interesting to note that the corresponding problem for the usual symmetric Bernoulli walk or
for Brownian motion remains open.
Received: 17 April 1998 相似文献
17.
Self-decomposable distributions are given as limits of normalized sums of independent random variables. We define semi-selfdecomposable
distributions as limits of subsequences of normalized sums. More generally, we introduce a way of making a new class of limiting
distributions derived from a class of distributions by taking the limits through subsequences of normalized sums, and define
the class of semi-selfdecomposable distributions and a decreasing sequence of subclasses of it. We give two kinds of necessary
and sufficient conditions for distributions belonging to those classes, one is in terms of the decomposability of random variables
and another is in terms of Lévy measures.
Received: 1 May 1997 / Revised version: 5 February 1998 相似文献
18.
Summary. We consider a continuous model for transverse magnetization of spins diffusing in a homogeneous Gaussian random longitudinal
field , where is the coupling constant giving the intensity of the random field. In this setting, the transverse magnetization is given
by the formula , where is the standard process of Brownian motion and is the covariance function of the original random field . We use large deviation techniques to show that the limit exists. We also determine the small behavior of the rate and show that it is indeed decaying as conjectured in the physics literature.
Received: 30 June 1995 / In revised form: 26 January 1996 相似文献
19.
20.
René L. Schilling 《Probability Theory and Related Fields》1998,112(4):565-611
Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C
c
∞(ℝ
n
)⊂D(A) and A|C
c
∞(ℝ
n
) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|∞≤c(1+|ξ|2) and |Imp(x,ξ)|≤c
0Rep(x,ξ). We show that the associated Feller process {X
t
}
t
≥0 on ℝ
n
is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour
of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β∞
x
:={λ>0:lim
|ξ|→∞
|
x
−
y
|≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ∞
x
:={λ>0:liminf
|ξ|→∞
|
x
−
y
|≤2/|ξ|
|ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙ
x
) that lim
t
→0
t
−1/λ
s
≤
t
|X
s
−x|=0 or ∞ according to λ>β∞
x
or λ<δ∞
x
. Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27].
Received: 21 July 1997 / Revised version: 26 January 1998 相似文献