A non-integral-dimensional random walk

We present a space-homogeneous, time-inhomogeneous random walk that behaves as if it were a simple random walk ind dimensions, whered is not necessarily an integer. Analogues of the Local Central Limit Theorem, Zero-One Laws, distance, angle, asymptotics on the Green's function and the hitting probability, recurrence and transience, and results about the intersection behavior of the random walk paths are obtained.     

Gap-free compositions and gap-free samples of geometric random variables

We study the asymptotic probability that a random composition of an integer n is gap-free, that is, that the sizes of parts in the composition form an interval. We show that this problem is closely related to the study of the probability that a sample of independent, identically distributed random variables with a geometric distribution is likewise gap-free.     

Quenched tail estimate for the random walk in random scenery and in random layered conductance

We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time asymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure.     

Efficient sampling of random permutations

We show how to uniformly distribute data at random (not to be confounded with permutation routing) in two settings that are able to deal with massive data: coarse grained parallelism and external memory. In contrast to previously known work for parallel setups, our method is able to fulfill the three criteria of uniformity, work-optimality and balance among the processors simultaneously. To guarantee the uniformity we investigate the matrix of communication requests between the processors. We show that its distribution is a generalization of the multivariate hypergeometric distribution and we give algorithms to sample it efficiently in the two settings.     

Coherent random permutations with biased record statistics

We consider random permutations that are defined coherently for all values of n, and for each n have a probability distribution which is conditionally uniform given the set of upper and lower record values. Our central example is a two-parameter family of random permutations that are conditionally uniform given the counts of upper and lower records. This family may be seen as an interpolation between two versions of Ewens’ distribution. We discuss characterisations of the conditionally uniform permutations, their asymptotic properties, constructions and relations to random compositions.     

Convexity of chance constraints with independent random variables

We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.     

The probability density function to the random linear transport equation

We present a formula to calculate the probability density function of the solution of the random linear transport equation in terms of the density functions of the velocity and the initial condition. We also present an expression for the joint probability density function of the solution in two different points. Our results have shown good agreement with Monte Carlo simulations.     

Exact transient analysis of a planar random motion with three directions

Consider a planar random motion with constant velocity and three directions forming the angles ~ /6, 5 ~ /6 and 3 ~ /2 with the x -axis, such that the random times between consecutive changes of direction perform an alternating renewal process. We obtain the probability law of the bidimensional stochastic process which describes location and direction of the motion. In the Markovian case when the random times between consecutive changes of direction are exponentially distributed, the transition densities of the motion are explicitly given. These are expressed in term of a suitable modified two-index Bessel function.     

Empirical processes and random projections

In this note, we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same L₂ norm. We present several geometric applications of this result, the most important of which is a sharpening of the Johnson-Lindenstrauss embedding Lemma. Our results apply to a large class of random matrices, as we only require that the matrix entries have a subgaussian tail.     

Continuous functions minimization by dynamic random search technique

Random search technique is the simplest one of the heuristic algorithms. It is stated in the literature that the probability of finding global minimum is equal to 1 by using the basic random search technique, but it takes too much time to reach the global minimum. Improving the basic random search technique may decrease the solution time. In this study, in order to obtain the global minimum fastly, a new random search algorithm is suggested. This algorithm is called as the Dynamic Random Search Technique (DRASET). DRASET consists of two phases, which are general search and local search based on general solution. Knowledge related to the best solution found in the process of general search is kept and then that knowledge is used as initial value of local search. DRASET’s performance was experimented with 15 test problems and satisfactory results were obtained.     

Multiscale analysis of exit distributions for random walks in random environments

We present a multiscale analysis for the exit measures from large balls in , of random walks in certain i.i.d. random environments which are small perturbations of the fixed environment corresponding to simple random walk. Our main assumption is an isotropy assumption on the law of the environment, introduced by Bricmont and Kupiainen. Under this assumption, we prove that the exit measure of the random walk in a random environment from a large ball, approaches the exit measure of a simple random walk from the same ball, in the sense that the variational distance between smoothed versions of these measures converges to zero. We also prove the transience of the random walk in random environment. The analysis is based on propagating estimates on the variational distance between the exit measure of the random walk in random environment and that of simple random walk, in addition to estimates on the variational distance between smoothed versions of these quantities. Partially supported by NSF grant DMS-0503775.     

Jensen's inequality for random elements in metric spaces and some applications

Jensen's inequality is extended to metric spaces endowed with a convex combination operation. Applications include a dominated convergence theorem for both random elements and random sets, a monotone convergence theorem for random sets, and other results on set-valued expectations in metric spaces and on random probability measures.     

Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

Summary We consider a model of random walk on ℤ^ν, ν≥2, in a dynamical random environment described by a field ξ={ξ _t (x): (t,x)∈ℤ^ν+1}. The random walk transition probabilities are taken as P(X _{t +1}= y|X _t = x,ξ _t =η) =P ₀( y−x)+ c(y−x;η(x)). We assume that the variables {ξ _t (x):(t,x) ∈ℤ^ν+1} are i.i.d., that both P ₀(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P ₀, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X _t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X _t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L ^p estimates for a class of functionals of the field. Received: 4 January 1996/In revised form: 26 May 1997     

Statistical maps and generalized random walks

We continue our study of statistical maps (equivalently, fuzzy random variables in the sense of Gudder and Bugajski). In the realm of fuzzy probability theory, statistical maps describe the transportation of probability measures on one measurable space into probability measures on another measurable space. We show that for discrete probability spaces each statistical map can be represented via a special matrix the rows of which are probability functions related to conditional probabilities and the columns are related to fuzzy n-partitions of the domain. Discrete statistical maps sending a probability measure p to a probability measure q can be represented via conditional distributions and correspond to joint probabilities on the product. The composition of statistical maps provide a tool to describe and to study generalized random walks and Markov chains.     

Occupation measure for random walk on the circle

We give bounds on the probability of deviation of the occupation measure of an interval on the circle for random walk.     

Collapse of attractors for ODEs under small random perturbations

This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to ‘collapse’. Unlike existing results of this type, no order preserving property is necessary.      

Random walk on the random connection model

We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like ${|x?y|}^{?\alpha }$ with $\alpha >d$, where $d$ denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits similar properties as classical discrete long-range percolation models studied by Berger (2002) with regard to recurrence and transience of the random walk. Moreover, we address a question which is related to a conjecture by Heydenreich, Hulshof and Jorritsma (2017) for this graph.     

Attainable densities for random maps

We consider a random map T=T(Γ,ω), where Γ=(τ₁,τ₂,…,τK) is a collection of maps of an interval and ω=(p₁,p₂,…,pK) is a collection of the corresponding position dependent probabilities, that is, pk(x)?0 for k=1,2,…,K and . At each step, the random map T moves the point x to τk(x) with probability pk(x). For a fixed collection of maps Γ, T can have many different invariant probability density functions, depending on the choice of the (weighting) probabilities ω. Most of the results in this paper concern random maps where Γ is a family of piecewise linear semi-Markov maps. We investigate properties of the set of invariant probability density functions of T that are attainable by allowing the probabilities in ω to vary in a certain class of functions. We prove that the set of all attainable densities can be determined algorithmically. We also study the duality between random maps generated by transformations and random maps constructed from a collection of their inverse branches. Such representation may be of greater interest in view of new methods of computing entropy [W. S?omczyński, J. Kwapień, K. ?yczkowski, Entropy computing via integration over fractal measures, Chaos 10 (2000) 180-188].     

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of “second order freeness” and interpret the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.     

Asymptotic properties of digit sequences of random numbers

We consider several aspects of the relationship between a [0, 1]‐valued random variable X and the random sequence of digits given by its m‐ary expansion. We present results for three cases: (a) independent and identically distributed digit sequences; (b) random variables X with smooth densities; (c) stationary digit sequences. In the case of i.i.d. an integral limit thorem is proved which applies for example to relative frequencies, yielding asymptotic moment identities. We deal with occurrence probabilities of digit groups in the case that X has an analytic Lebesgue density. In the case of stationary digits we determine the distribution of X in terms of their transition functions. We study an associated [0, 1]‐valued Markov chain, in particular its ergodicity, and give conditions for the existence of stationary digit sequences with prespecified transition functions. It is shown that all probability measures induced on [0, 1] by such sequences are purely singular except for the uniform distribution. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)