Relative Complexity of Random Walks in Random Scenery in the absence of a weak invariance principle for the local times

@article{Deligiannidis2015RelativeCO,
  title={Relative Complexity of Random Walks in Random Scenery in the absence of a weak invariance principle for the local times},
  author={George Deligiannidis and Zemer Kosloff},
  journal={arXiv: Probability},
  year={2015}
}
We answer the question of Aaronson about the relative complexity of Random Walks in Random Sceneries driven by either aperiodic two dimensional random walks, two-dimensional Simple Random walk, or by aperiodic random walks in the domain of attraction of the Cauchy distribution. A key step is proving that the range of the random walk satisfies the F\"olner property almost surely. 
The boundary of the range of a random walk and the Følner property
The range process Rn of a random walk is the collection of sites visited by the random walk up to time n. In this paper we deal with the question of whether the range process of a random walk or the
On the quenched functional CLT in 2-d random sceneries, examples
We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random eld (r.f.) along a 2d-random walk in dierent situations: when the r.f. is iid with a second order moment
Boundary of the Range of a random walk and the F\"olner property
The ranges process $R_n$ of a random walk is the collection of sites visited by the random walk up to time $n$. In this work we deal with the question of whether the range process of a random walk or
Law of large numbers for the drift of two-dimensional wreath product
We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $(2+\epsilon)$-moment. This result is in
Optimal bounds for self-intersection local times
For a random walk $S_n, n\geq 0$ in $\mathbb{Z}^d$, let $l(n,x)$ be its local time at the site $x\in \mathbb{Z}^d$. Define the $\alpha$-fold self intersection local time $L_n(\alpha) := \sum_{x}

References

SHOWING 1-10 OF 31 REFERENCES
Relative Complexity of random walks in random sceneries
Relative complexity measures the complexity of a probability preserving transformation relative to a factor being a sequence of random variables whose exponential growth rate is the relative entropy
Entropy andσ-algebra equivalence of certain random walks on random sceneries
LetX=X0,X1,…be a stationary sequence of random variables defining a sequence space Σ with shift mapσ and let (Tt, Ω) be an ergodic flow. Then the endomorphismTX(x, ω)=(σ(x),Tx0(ω)) is known as a
Intersections of random walks
We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk.
The Range of Stable Random Walks
Limit theorems are proved for the range of d-dimensional random walks in the domain of attraction of a stable process of index P3. The range Rn is the number of distinct sites of Zd visited by the
Scenery entropy as an invariant of RWRS processes
Probabilistic models of random walks in random sceneries give rise to examples of probability-preserving dynamical systems. A point in the state spaces consists of a walk-trajectory and a scenery,
Random Walk: A Modern Introduction
TLDR
This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice and is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.
A central limit theorem for two-dimensional random walks in random sceneries
Let $S_n$, $n\in\bold N$, be a recurrent random walk on ${\bold Z}^2$ $(S_0=0)$ and let $\xi(\alpha)$, $\alpha\in{\bold Z}^2$, be i.i.d. $\bold R$-valued centered random variables. It is shown that
Mixing properties of the generalized T, T-1-process
Consider a general random walk on ℤd together with an i.i.d. random coloring of ℤd. TheT, T-1-process is the one where time is indexed by ℤ, and at each unit of time we see the step taken by the walk
Measure-theoretic complexity of ergodic systems
AbstractWe define an invariant of measure-theoretic isomorphism for dynamical systems, as the growth rate inn of the number of small $$\bar d$$ -balls aroundα-n-names necessary to cover most of the
...
1
2
3
4
...