Discrete-time Random Walk Research Articles

Uniform Convergence of Martingales in the Branching Random Walk

In a discrete-time supercritical branching random walk, let $Z^{(n)}$ be the point process formed by the $n$th generation. Let $m(\lambda)$ be the Laplace transform of the intensity measure of $Z^{(1)}$. Then $W^{(n)}(\lambda) = \int e^{-\lambda x}Z^{(n)}(dx)/m(\lambda)^n$, which is the Laplace transform of $Z^{(n)}$ normalized by its expected value, forms a martingale for any $\lambda$ with $|m(\lambda)|$ finite but nonzero. The convergence of these martingales uniformly in $\lambda$, for $\lambda$ lying in a suitable set, is the first main result of this paper. This will imply that, on that set, the martingale limit $W(\lambda)$ is actually an analytic function of $\lambda$. The uniform convergence results are used to obtain extensions of known results on the growth of $Z^{(n)}(nc + D)$ with $n$, for bounded intervals $D$ and fixed $c$. This forms the second part of the paper, where local large deviation results for $Z^{(n)}$ which are uniform in $c$ are considered. Finally, similar results, both on martingale convergence and uniform local large deviations, are also obtained for continuous-time models including branching Brownian motion.

A note on the distribution of response times for a random walk with Gaussian increments

In this note, response time distribution predictions for the two-absorbing-barrier Gaussian-increments random walk in discrete time are derived. This derivation, which neglects the excess over the absorbing barriers on termination, involves an implicit limiting operation on the interval between increments to the walk, thereby approximating the discrete time random walk with a continuous time Wiener diffusion process. Two simple procedures for correcting for the excess over the absorbing barriers are considered, which increase the barriers by an estimate of the expected value of the excess. These corrections substantially improve the accuracy of the predictions, as shown by the fit between theoretical and simulated distributions. One implication of this result is that it is unlikely to be possible to distinguish between discrete and continuous sampling models on the basis of goodness of fit indices alone.

Master equation versus random walk approaches to phenomena far from equilibrium

We study the relation between (discrete time) random walk and (continuous time) master equation approaches to phenomena in non-equilibrium systems. The equivalence of these approaches is proved under very broad conditions. The obtained equivalence theorem includes generalized master equations that can be derived from quantum mechanics in an exact manner. Results are illustrated by and applied to examples from nuclear physics. A certain soluble model example, concerning the Z and N distributions of emergent fragments in heavy-ion collisions, is presented in order to re-interpret earlier results of Hiller et al. seemingly contradicting the equivalence theorem. It is furthermore shown that experimental observables in nuclear physics applications are sensitive only to the mean time between intranuclear collisions, and not to the nature of the distribution of waiting times between these events.

Stochastic models for the completeness of stratigraphic sections

Properties of stratigraphic completeness are determined here from a Brownian motion model of sediment accumulation. This avoids flaws inherent in application of a discrete-time random walk to the time span, rather than thickness, of sediment layers. Both discrete and continuous models show that the concept of stratigraphic completeness is meaningful only when the time scale is specified. From the discrete model, not surprisingly, completeness improves with increasing relative frequency and average thickness of depositional increments and the error of completeness estimation should decrease for longer sections. The continuous model shows that two dimensionless products determine the probability that a given time interval will be recorded by some preserved sediment. The first is the ratio of the age of the interval to its time span; the second is the product of the square root of the time span and ratio of the mean to the standard deviation of accumulation rate. Expected completeness is the average of these probabilities for all successive intervals of the given time span. For long sections, completeness may be estimated from the second dimensionless product alone. The two dimensionless products are sufficient to predict the relationship of accumulation rate to time span, the distribution of bed thickness, and the weak association of completeness and section thickness.

On the range of a constrained random walk

We study statistical properties of the range (= number of distinct sites visited) of a lattice random walk in discrete time constrained to visit a given site at a given time. In particular, we calculate the mean and obtain a bound on the variance of the range in the large time limit. The results are applied to a problem involving an unconstrained random walk in the presence of randomly distributed traps. A key role is played by the associated random walk that is obtained from the original random walk via a Cramer transform.

On the range of a constrained random walk

We study statistical properties of the range (= number of distinct sites visited) of a lattice random walk in discrete time constrained to visit a given site at a given time. In particular, we calculate the mean and obtain a bound on the variance of the range in the large time limit. The results are applied to a problem involving an unconstrained random walk in the presence of randomly distributed traps. A key role is played by the associated random walk that is obtained from the original random walk via a Cramer transform.

Discrete-time random walks on diagrams (graphs) with cycles.

After a review of the diagram method for continuous-time random walks on graphs with cycles, the method is extended to discrete-time random walks. The basic theorems carry over formally from continuous time to discrete time. Three problems in tennis probabilities are used to illustrate random walks on discrete-time diagrams with cycles.

The passage from random walk to diffusion in quantum probability

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

The passage from random walk to diffusion in quantum probability

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

Continuous time random walks with momentless waiting time distributions

Continuous time random walk (CTRW) is studied in the case of momentless waiting time distributions with long-time tail t-α, 0<α<1. It is found that the hopping event set is a well defined random fractal whose fractal (Hausdorff) dimension is α, and if one quantity of discrete time random walk R(n) [e.g., S(n) the number of distinct sites visited up to step n] has scaling exponent β>0, then its corresponding quantity of CTRW R(T) (e.g., S(t) the number of distinct sites visited up to time t has scaling exponent αβ.

Consistent Estimation in Partially Observed Random Walks

If a discrete-time random walk, consisting of the sum of independent increments from an unknown underlying distribution, is observed at every time instant then it is clear that the underlying distribution can be consistently estimated. If, however, we are restricted to observing only a subset of the times, and if this subset is too sparse, then a central limit effect takes over and only two moments can be consistently estimated. We show that divergence of the sum of the reciprocals of the observation time-intervals is a necessary and sufficient condition to permit consistent estimation of the third moment. Corresponding conditions permit consistent estimation of moments of higher order. An explicit consistent estimator for the distribution itself is presented when all moments can be consistently estimated and the distribution is determined by its moments.

Asymptotic properties of multistate random walks. I. Theory

A calculation is presented of the long-time behavior of various random walk properties (moments, probability of return to the origin, expected number of distinct sites visited) for multistate random walks on periodic lattices. In particular, we consider inhomogeneous periodic lattices, consisting of a periodically repeated unit cell which contains a finite number of internal states (sites). The results are identical to those for perfect lattices except for a renormalization of coefficients. For walks without drift, it is found that all the asymptotic random walk properties are determined by the diffusion coefficients for the multistate random walk. The diffusion coefficients can be obtained by a simple matrix algorithm presented here. Both discrete and continuous time random walks are considered. The results are not restricted to nearest-neighbor random walks but apply as long as the single-step probability distributions associated with each of the internal states have finite means and variances.

M/M/1 Queueing Decision Processes with Monotone Hysteretic Optimal Policies

This paper studies an M/M/1 queueing decision process in which the arrival and service rates are chosen from a finite set whenever the queue length changes. Each choice has a switching cost depending on the chosen rates and those currently in use, and a usage cost depending on the chosen rates and the queue length; these switching and usage cost functions are submodular and satisfy additional technical conditions. The system objective is to find a policy for dynamically choosing the rates, based on the current rates and queue length, that minimizes the expected total discounted cost or average cost over an infinite horizon. We prove that there is a monotone hysteretic optimal policy in which the arrival and service rates are decreasing and increasing, respectively, in the queue length; there is a hysteresis (retardation) in the changing of the actions due to the switching costs. We establish this result by showing that such optimal policies exist for an equivalent discrete-time random walk decision process. Our results confirm a decade-old conjecture that the M/M/1 queue has a monotone hysteretic optimal control policy.

Fractal and lacunary stochastic processes

Discrete-time random walks simulate diffusion if the single-step probability density function (jump distribution) generating the walk is sufficiently shortranged. In contrast, walks with long-ranged jump distributions considered in this paper simulate Levy or stable processes. A one-dimensional walk with a selfsimilar jump distribution (the Weierstrass random walk) and its higherdimensional generalizations generate fractal trajectories if certain transience criteria are met and lead to simple analogs of deep results on the Hausdorff-Besicovitch dimension of stable processes. The Weierstrass random walk is lacunary (has gaps in the set of allowed steps) and its characteristic function is Weierstrass' non-differentiable function. Other lacunary random walks with characteristic functions related to Riemann's zeta function and certain numbertheoretic functions have very interesting analytic structure.

A First Passage Problem in a Random Walk with a Quality Control Application

SUMMARY This paper is concerned with a first passage problem for a discrete time random walk which arises in a quality control scheme proposed by the author. There are two barriers, one of which is absorbing and the other such that the random walk particle is immediately returned to the origin upon reaching it. An explicit expression is derived for the probability distribution of the time to absorption when the barriers are symmetrically placed on either side of the origin.

First-Passage Time for Oscillators With Nonlinear Damping

A simple numerical scheme is proposed for computing the probability of first passage failure, P(T), in an interval O-T, for oscillators with nonlinear damping. The method depends on the fact that, when the damping is light, the amplitude envelope, A(t), can be accurately approximated as a one-dimensional Markov process. Hence, estimates of P(T) are found, for both single and double-sided barriers, by solving the Fokker-Planck equation for A(t) with an appropriate absorbing barrier. The numerical solution of the Fokker-Planck equation is greatly simplified by using a discrete time random walk analog of A(t), with appropriate statistical properties. Results obtained by this method are compared with corresponding digital simulation estimates, in typical cases.

Cerebral embolism as a complication of coronary thrombosis: Dozzi, Daniel L.: Am. J. M. Sc. 194: 824, 1937

In the discrete-time fractional random walk model a market with one risky asset affords an arbitrage opportunity as described by Cutland et al. [Cutland NJ, Kopp PE, Willinger W. Stock price returns and the Joseph effect: a fractional version of the Black–Scholes model. In: Russo Francesco, Bolthausen Erwin, Dozzi Marco, editors. Seminar on 6 stochastic analysis, random fields and applications, pp. 327–351. Seminar on stochastic analysis, random fields and applications. Ascona: Centro Stefano Franscini; 1993, Progress in probability 36. Birkhauser Verlag; 1995.] and Sottinen [Sottinen Tommi. Fractional Brownian motion, random walks and binary market models. Finance Stoch 2001;5(3):343–355]. We briefly discuss these results and compute a numerical example in a fractional binomial model as illustration and mention an option pricing model for assets the returns of which are driven by a fractional Brownian motion [Yaozhong Hu, Bernt Øksendal. Fractional white noise calculus and applications to finance. Infin Dimens Anal Quant Probability Rel Top 2003;6:1–32, ISSN 0219-0257; Fajardo J, Cajueiro DO. Volatility estimation and option pricing with fractional Brownian motion, October 2003. Available from: http://ideas.repec.org/p/ibm/finlab/flwp53.html].