Rank-based selection strategies for the random walk process

In many decision situations such as hiring a secretary, selling an asset, or seeking a job, the value of each offer, applicant, or choice is assumed to be an independent, identically distributed random variable. In this paper, we consider a special case where the observations are auto-correlated as in the random walk model for stock prices. For a given random walk process of n observations, we explicitly compute the probability that the j-th observation in the sequence is the maximum or minimum among all n observations. Based on the probability distribution of the rank, we derive several distribution-free selection strategies under which the decision maker's expected utility of selecting the best choice is maximized. We show that, unlike in the classical secretary problem, evaluating more choices in the random walk process does not increase the likelihood of successfully selecting the best.     

On the Wick theorem for mixtures of centered Gaussian distributions

We consider centered conditionally Gaussian d-dimensional vectors X with random covariance matrix Ξ having an arbitrary probability distribution law on the set of nonnegative definite symmetric d × d matrices M _d ⁺. The paper deals with the evaluation problem of mean values \( E\left[ {\prod\nolimits_{i = 1}^{2n} {\left( {{c_i},X} \right)} } \right] \) for c _i ∈ ? ^d , i = 1, …, 2n, extending the Wick theorem for a wide class of non-Gaussian distributions. We discuss in more detail the cases where the probability law ?(Ξ) is infinitely divisible, the Wishart distribution, or the inverse Wishart distribution. An example with Ξ \( = \sum\nolimits_{j = 1}^m {{Z_j}{\sum_j}} \), where random variables Z _j , j = 1, …, m, are nonnegative, and Σ _j ∈ M _d ⁺, j = 1, …, m, are fixed, includes recent results from Vignat and Bhatnagar, 2008.     

Convergence of Point Processes with Weakly Dependent Points

For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}}For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process N_n=?_j=1ⁿd_Xj,nN_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}} to an infinitely divisible point process. From the point process convergence we obtain the convergence in distribution of the partial sum sequence S _n =∑ _j=1 ⁿ X _j,n to an infinitely divisible random variable whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.     

Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

The oscillating random walk

{Y_n;n=0, 1, …} denotes a stationary Markov chain taking values in R^d. As long as the process stays on the same side of a fixed hyperplane E⁰, it behaves as an ordinary random walk with jump measure μ or ν, respectively. Thus ordinary random walk would be the special case μ = ν. Also the process Y′_n = |Y′_n?1?Z_n| (with the Z_n as i.i.d. real random varia bles) may be regarded as a special case. The general process is studied by a Wiener–Hopf type method. Exact formulae are obtained for many quantities of interest. For the special case that the Y_n are integral-valued, renewal type conditions are established which are necessary and sufficient for recurrence.     

Local asymptotics for the time of first return to the origin of transient random walk

We consider a transient random walk on Z^d which is asymptotically stable, without centering, in a sense which allows different norming for each component. The paper is devoted to the asymptotics of the probability of the first return to the origin of such a random walk at time n.     

Level crossing probabilities II: Polygonal recurrence of multidimensional random walks

In part I we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is O(n−1/2). In higher dimensions we call a random walk ‘polygonally recurrent’ if there is a bounded set, hit by infinitely many of the straight lines between two consecutive sites a.s. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z³ in the sense of Mauldin, Monticino and von Weizsäcker [R.D. Mauldin, M. Monticino, H. von Weizsäcker, Directionally reinforced random walks, Adv. Math. 117 (1996) 239-252] is transient. On the other hand, we construct an example of a transient but polygonally recurrent random walk with independent components on Z².     

On Large Deviations of Multivariate Heavy-Tailed Random Walks

Let {S _n ;n=1,2,…} be a random walk in R ^d and E(S ₁)=(μ ₁,…,μ _d ). Let a _j >μ _j for j=1,…,d and A=(a ₁,∞)×⋅⋅⋅×(a _d ,∞). We are interested in the probability P(S _n /n∈A) for large n in the case where the components of S ₁ are heavy tailed. An objective is to associate an exact power with the aforementioned probability. We also derive sharper asymptotic bounds for the probability and show that in essence, the occurrence of the event {S _n /n∈A} is caused by large single increments of the components in a specific way.      

Random matrix models of stochastic integral type for free infinitely divisible distributions

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ?? ₀ ^?? e ^?1 d?? _t ^d , d ?? 1, where ?? _t ^d is a d × d matrix-valued Lévy process satisfying an I _log condition.     

Distinguishing sceneries by observing the scenery along a random walk path

LetG be an infinite connected graph with vertex setV. Ascenery onG is a map ξ :V → 0, 1 (equivalently, an assignment of zeroes and ones to the vertices ofG). LetS _{n n≥0} be a simple random walk onG, starting at some distinguished vertex v₀. Now let ξ and η be twoknown sceneries and assume that we observe one of the two sequences ξ(S _n) _n≥0 or {η(S _n)} _n≥0 but we do not know which of the two sequences is observed. Can we decide, with a zero probability of error, which of the two sequences is observed? We show that ifG = Z orG = Z², then the answer is “yes” for each fixed ξ and “almost all” η. We also give some examples of graphsG for which almost all pairs (ξ, η) are not distinguishable, and discuss some variants of this problem.     

Annealed deviations of random walk in random scenery

Let (Zn)_n∈N be a d-dimensional random walk in random scenery, i.e., with (Sk)_k∈N0 a random walk in Zd and (Y(z))_z∈Zd an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of for various choices of sequences n(bn) in [1,∞). Depending on n(bn) and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497-527] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].     

Bi-criteria scheduling problems: Number of tardy jobs and maximum weighted tardiness

Consider a single machine and a set of n jobs that are available for processing at time 0. Job j has a processing time p_j, a due date d_j and a weight w_j. We consider bi-criteria scheduling problems involving the maximum weighted tardiness and the number of tardy jobs. We give NP-hardness proofs for the scheduling problems when either one of the two criteria is the primary criterion and the other one is the secondary criterion. These results answer two open questions posed by Lee and Vairaktarakis in 1993. We consider complexity relationships between the various problems, give polynomial-time algorithms for some special cases, and propose fast heuristics for the general case. The effectiveness of the heuristics is measured by empirical study. Our results show that one heuristic performs extremely well compared to optimal solutions.     

On the average of a random walk

Let {S_n, n ϵ N)} be a simple random walk and denote by A_n its time average: A_n = (S₁+ …+S_n)/n. We give an integral test for the lower bound on A_n, thus giving an affirmative answer to a conjecture of P. Erdös (private communication) that A_n will return to a fixed region around the origin infinitely often with probability 1 in 1 dimension whereas in 2 or more dimensions it will return only finitely many times.     

Balanced allocation on graphs: A random walk approach

We propose algorithms for allocating n sequential balls into n bins that are interconnected as a d‐regular n‐vertex graph G, where d ≥ 3 can be any integer. In general, the algorithms proceeds in n succeeding rounds. Let ? > 0 be an integer, which is given as an input to the algorithms. In each round, ball 1 ≤ t ≤ n picks a node of G uniformly at random and performs a nonbacktracking random walk of length ? from the chosen node and simultaneously collects the load information of a subset of the visited nodes. It then allocates itself to one of them with the minimum load (ties are broken uniformly at random). For graphs with sufficiently large girths, we obtain upper and lower bounds for the maximum number of balls at any bin after allocating all n balls in terms of ?, with high probability.     

A rank-based approach to the sequential selection and assignment problem

In the classical sequential assignment problem, “machines” are to be allocated sequentially to “jobs” so as to maximize the expected total return, where the return from an allocation of job j to machine k is the product of the value x_j of the job and the weight p_k of the machine. The set of m machines and their weights are given ahead of time, but n jobs arrive in sequential order and their values are usually treated as independent, identically distributed random variables from a known univariate probability distribution with known parameter values. In the paper, we consider a rank-based version of the sequential selection and assignment problem that minimizes the sum of weighted ranks of jobs and machines. The so-called “secretary problem” is shown to be a special case of our sequential assignment problem (i.e., m = 1). Due to its distribution-free property, our rank-based assignment strategy can be successfully applied to various managerial decision problems such as machine scheduling, job interview, kidney allocations for transplant, and emergency evacuation plan of patients in a mass-casualty situation.     

Local Particles Numbers in Critical Branching Random Walk

Critical catalytic branching random walk on an integer lattice ? ^d is investigated for all d∈?. The branching may occur at the origin only and the start point is arbitrary. The asymptotic behavior, as time grows to infinity, is determined for the mean local particles numbers. The same problem is solved for the probability of the presence of particles at a fixed lattice point. Moreover, the Yaglom type limit theorem is established for the local number of particles. Our analysis involves construction of an auxiliary Bellman–Harris branching process with six types of particles. The proofs employ the asymptotic properties of the (improper) c.d.f. of hitting times with taboo. The latter notion was recently introduced by the author for a non-branching random walk on ? ^d .     

The mean number of sites visited by a pinned random walk

This paper concerns the number Z _n of sites visited up to time n by a random walk S _n having zero mean and moving on the d-dimensional square lattice Z ^d . Asymptotic evaluation of the conditional expectation of Z _n given that S ₀ = 0 and S _n = x is carried out under 2 + δ moment conditions (0 ≤ δ ≤ 2) in the cases d = 2, 3. It gives an explicit form of the leading term and reasonable estimates of the remainder term (depending on δ) valid uniformly in each parabolic region of (x, n). In the case x = 0 the problem has been studied for the simple random walk and its analogue for Brownian motion; the estimates obtained here are finer than or comparable to those found in previous works. Supported in part by Monbukagakusho grand-in-aid no. 15540109.     

Asymptotic behavior of a stochastic growth process associated with a system of interacting branching random walksComportement asymptotique d'un processus stochastique de croissance associé à un système de marches aléatoires en interaction

We study a continuous time growth process on $\text{Z}{}^{\mathrm{d}}$ (d?1) associated to the following interacting particle system: initially there is only one simple symmetric continuous time random walk of total jump rate one located at the origin; then, whenever a random walk visits a site still unvisited by any other random walk, it creates a new independent random walk starting from that site. Let us call P_d the law of such a process and S⁰_d(t) the set of sites, visited by all walks by time t. We prove that there exists a bounded, non-empty, convex set $\text{C}{}_{\mathrm{d}}\text{?}\text{R}{}^{\mathrm{d}}$, such that for every ε>0, P_d-a.s. eventually in t, the set S_d⁰(t) is within an ε neighborhood of the set [C_dt], where for $\text{A?}\text{R}{}^{\mathrm{d}}$ we define $\text{[A]:=A∩}\text{Z}{}^{\mathrm{d}}$. Moreover, for d large enough, the set C_d is not a ball under the Euclidean norm. We also show that the empirical density of particles within S_d⁰(t) converges weakly to a product Poisson measure of parameter one. To cite this article: A.F. Ram??rez, V. Sidoravicius, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 821–826.     

Revisits for transient random walk

For a random walk on the integers define R_n as the number of (distinct) states visited in the first n steps and Z_n as the number of states visited in the first n steps which are never revisited. Here we deal with transient walks. The increments of Z_n form a stationary process and various central limit results and an iterated logarithm result are obtained for Z_n from known results on stationary processes. Furthermore, the limit behaviour of R_n is closely related to that of Z_n; this relationship is elucidated and corresponding limit results for R_n are then read off from those for Z_n.     

Integers divisible by sums of powers of their prime factors

For each positive integer j, let βj(n):=_∑p|npj. Given a fixed positive integer k, we show that there are infinitely many positive integers n having at least two distinct prime factors and such that βj(n)|n for each j∈{1,2,…,k}.