A note on the random mean ergodic theorem

2011

I discuss the impact various papers have had on my own work. Date: January 2011. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary:

2017

In this paper, we study dynamical properties as hypercyclicity, supercyclicity, frequent hypercyclicity and chaoticity for transition operators associated to countable irreductible Markov chains. As particular cases, we consider simple random walks on Z and Z+.

2012

We consider random walks in a balanced random environment in Z d , d ≥ 2. We first prove an invariance principle (for d ≥ 2) and the transience of the random walks when d ≥ 3 (recurrence when d = 2) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.

2001

Main laws of probability theory, when applied to individual sequences, have a "robustness" property under small violations of randomness. For example, the law of large numbers for the symmetric Bernoulli scheme holds for a sequence where the randomness deficiency of its initial fragment of length n grows as o(n). The law of iterated logarithm holds if the randomness deficiency grows as o(log log n). We prove that Birkhoff's individual ergodic theorem is nonrobust in this sense. If the randomness deficiency grows arbitrarily slowly on initial fragments of an infinite sequence, this theorem can be violated. An analogous nonrobustness property holds for the Shannon-McMillan-Breiman theorem.

2016

Two results on Asymptotic Behaviour of Random Walks in Random Environment

arXiv (Cornell University), 2010

In this paper, we consider random walk in random environment on Z d (d ≥ 1) and prove the Strassen's strong invariance principle for this model, via martingale argument and the theory of fractional coboundaries of Derriennic and Lin [4], under some conditions which require the variance of the quenched mean has a subdiffusive bound. The results partially fill the gaps between law of large numbers and central limit theorems.

Theoretical Computer Science, 1998

In the framework of the Kolmogorov's approach to the substantiation of the probability theory and information theory on the base of the theory of algorithms we try to formulate probabilistic laws, i.e. statements of the form P{w]A(w)} = I, where A(o)) is some formula, in "pointwise" form "if w is random then A(w) holds". Nevertheless, not all proofs of such laws can be directly translated into the algorithmic form. In [I l] two examples have been distinguished Birkhoff's [2] ergodic theorem and Shannon-McMillan-Breiman theorem of information theor) [I], In this paper an analysis of algorithmic effectiveness of these theorems is given. We prove that Birkhoff's ergodic theorem is indeed in some strong sense "nonconstructive". At the same time the claim to formulate probabilistic laws for algorithmically random sequences is not so restrictive. We present the versions of these laws for individual random sequences.

Studia Scientiarum Mathematicarum Hungarica, 2008

In this paper we will give a simple symmetric random walk analogue of Lévy’s Theorem. We will give a new definition of a local time of the simple symmetric random walk. We apply a discrete Itô formula to some absolute value like function to obtain a discrete Tanaka formula. Results in this paper rely upon a discrete Skorokhod reflection argument. This random walk analogue of Lévy’s theorem was already obtained by G. Simons ([14]) but it is still worth noting because we will use a discrete stochastic analysis to obtain it and this method is applicable to other research. We note some connection with previous results by Csáki, Révész, Csörgő and Szabados. Finally we observe that the discrete Lévy transformation in the present version is not ergodic. Lastly we give a Lévy-type theorem for simple nonsymmetric random walk using a discrete bang-bang process.

Journal of Physics: Conference Series, 2021

We study a strongly Non-Markovian variant of random walk in which the probability of visiting a given site i is a function f of number of previous visits v(i) to the site. If the probability is inversely proportional to number of visits to the site, say f(i)=1/(v(i)+1)α the probability distribution of visited sites tends to be flat for α>0 compared to simple random walk. For f(i)=exp(-v(i)), we observe a distribution with two peaks. The origin is no longer the most probable site. The probability is maximum at site k(t) which increases in time. For f(i)=exp(-v(i)) and for α>0 the properties do not change as the walk ages. However, for α<0, the properties are similar to simple random walk asymptotically. We study lattice covering time for these functions. The lattice covering time scales as Nz, with z=2, for α ≤ 0, z>2 for a >0 and z<2 for f(i)=exp(-v(i)).

Israel Journal of Mathematics, 2005

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f n } ⊂ L p , based on the size of the norms of sums of sub-blocks of the first n functions. The results are applied to the study of a.e. convergence of series n a n T n g n α when T is an L 2-contraction, g ∈ L 2 , and {a n } is an appropriate sequence. Given a sequence {f n } ⊂ L p (Ω, µ), 1 < p ≤ 2, of independent centered random variables, we study conditions for the existence of a set of x of µ-probability 1, such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random power series n f n (x)T n g converges π-a.e. The conditions are used to show that for {f n } centered i.i.d. with f 1 ∈ L log + L, there exists a set of x of full measure such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random series n fn(x)T n g n converges π-a.e.