1,312 research outputs found
Empirical processes for recurrent and transient random walks in random scenery
In this paper, we are interested in the asymptotic behaviour of the sequence
of processes with \begin{equation*}
W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{\xi_{S_k}\leq s\}}-s\big)
\end{equation*} where is a sequence of independent
random variables uniformly distributed on and
is a random walk evolving in , independent of the 's. In
Wendler (2016), the case where is a recurrent random
walk in such that converges in
distribution to a stable distribution of index , with ,
has been investigated. Here, we consider the cases where is either: a) a transient random walk in , b) a recurrent
random walk in such that
converges in distribution to a stable distribution of index
A functional approach for random walks in random sceneries
A functional approach for the study of the random walks in random sceneries
(RWRS) is proposed. Under fairly general assumptions on the random walk and on
the random scenery, functional limit theorems are proved. The method allows to
study separately the convergence of the walk and of the scenery: on the one
hand, a general criterion for the convergence of the local time of the walk is
provided, on the other hand, the convergence of the random measures associated
with the scenery is studied. This functional approach is robust enough to
recover many of the known results on RWRS as well as new ones, including the
case of many walkers evolving in the same scenery.Comment: 23
Large deviations for intersection local times in critical dimension
Let be a continuous time simple random walk on
(), and let be the time spent by on the site
up to time . We prove a large deviations principle for the -fold
self-intersection local time in the
critical case . When is integer, we obtain similar results
for the intersection local times of independent simple random walks.Comment: Published in at http://dx.doi.org/10.1214/09-AOP499 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Self-intersection local times of random walks: Exponential moments in subcritical dimensions
Fix , not necessarily integer, with . We study the -fold
self-intersection local time of a simple random walk on the lattice up
to time . This is the -norm of the vector of the walker's local times,
. We derive precise logarithmic asymptotics of the expectation of
for scales that are bounded from
above, possibly tending to zero. The speed is identified in terms of mixed
powers of and , and the precise rate is characterized in terms of
a variational formula, which is in close connection to the {\it
Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation
principle for for deviation functions satisfying
t r_t\gg\E[\|\ell_t\|_p]. Informally, it turns out that the random walk
homogeneously squeezes in a -dependent box with diameter of order to produce the required amount of self-intersections. Our main tool is
an upper bound for the joint density of the local times of the walk.Comment: 15 pages. To appear in Probability Theory and Related Fields. The
final publication is available at springerlink.co
A local limit theorem for random walks in random scenery and on randomly oriented lattices
International audienceRandom walks in random scenery are processes defined by , where and are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index and respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when and as , of , for some suitable depending on and . Here we are interested in the convergence, as , of , when x\in \RR is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results
Large deviations for self-intersection local times of stable random walks
Let be a random walk on . Let the local time at the state and the q-fold self-intersection local
time (SILT). In \cite{Castell} Castell proves a large deviations principle for
the SILT of the simple random walk in the critical case . In the
supercritical case , Chen and M\"orters obtain in \cite{ChenMorters}
a large deviations principle for the intersection of independent random
walks, and Asselah obtains in \cite{Asselah5} a large deviations principle for
the SILT with . We extend these results to an -stable process
(i.e. ) in the case where .Comment: 22 page
An asymptotic variance of the self-intersections of random walks
We present a Darboux-Wiener type lemma and apply it to obtain an exact
asymptotic for the variance of the self-intersection of one and two-dimensional
random walks. As a corollary, we obtain a central limit theorem for random walk
in random scenery conjectured by Kesten and Spitzer in 1979
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
Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes
In this paper we prove exact forms of large deviations for local times and
intersection local times of fractional Brownian motions and Riemann-Liouville
processes. We also show that a fractional Brownian motion and the related
Riemann-Liouville process behave like constant multiples of each other with
regard to large deviations for their local and intersection local times. As a
consequence of our large deviation estimates, we derive laws of iterated
logarithm for the corresponding local times. The key points of our methods: (1)
logarithmic superadditivity of a normalized sequence of moments of
exponentially randomized local time of a fractional Brownian motion; (2)
logarithmic subadditivity of a normalized sequence of moments of exponentially
randomized intersection local time of Riemann-Liouville processes; (3)
comparison of local and intersection local times based on embedding of a part
of a fractional Brownian motion into the reproducing kernel Hilbert space of
the Riemann-Liouville process.Comment: To appear in the Annals of Probabilit
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