Hitting and returning to rare events for all alpha-mixing processes

We prove that for any a-mixing stationary process the hitting time of any n-string A(n) converges, when suitably normalized, to an exponential law. We identify the normalization constant lambda(A(n)). A similar statement holds also for the return time. To establish this result we prove two other results of independent interest. First, we show a relation between the rescaled hitting time and the rescaled return time, generalizing a theorem of Haydn, Lacroix and Vaienti. Second, we show that for positive entropy systems, the probability of observing any n-string in n consecutive observations goes to zero as n goes to infinity. (c) 2010 Elsevier B.V. All rights reserved.

Subgeometric rates of convergence for a class of continuous-time Markov process

Let (Phi(t))(t is an element of R+) be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure pi. We investigate the rates of convergence of the transition function P-t(x, (.)) to pi; specifically, we find conditions under which r(t) vertical bar vertical bar P-t (x, (.)) - pi vertical bar vertical bar -> 0 as t -> infinity, for suitable subgeometric rate functions r(t), where vertical bar vertical bar - vertical bar vertical bar denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

On the Moving Boundary Hitting Probability for the Brownian Motion

2000 Mathematics Subject Classification: 60J65.

The collision branching process

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

Developing the follow-up schedules for women with breast cancer

Background: The follow-up care for women with breast cancer requires an understanding of disease recurrence patterns and the follow-up visit schedule should be determined according to the times when the recurrence are most likely to occur, so that preventive measure can be taken to avoid or minimize the recurrence. Objective: To model breast cancer recurrence through stochastic process with an aim to generate a hazard function for determining a follow-up schedule. Methods: We modeled the process of disease progression as the time transformed Weiner process and the first-hitting-time was used as an approximation of the true failure time. The women's "recurrence-free survival time" or a "not having the recurrence event" is modeled by the time it takes Weiner process to cross a threshold value which represents a woman experiences breast cancer recurrence event. We explored threshold regression model which takes account of covariates that contributed to the prognosis of breast cancer following development of the first-hitting time model. Using real data from SEER-Medicare, we proposed models of follow-up visits schedule on the basis of constant probability of disease recurrence between consecutive visits. Results: We demonstrated that the threshold regression based on first-hitting-time modeling approach can provide useful predictive information about breast cancer recurrence. Our results suggest the surveillance and follow-up schedule can be determined for women based on their prognostic factors such as tumor stage and others. Women with early stage of disease may be seen less frequently for follow-up visits than those women with locally advanced stages. Our results from SEER-Medicare data support the idea of risk-controlled follow-up strategies for groups of women. Conclusion: The methodology we proposed in this study allows one to determine individual follow-up scheduling based on a parametric hazard function that incorporates known prognostic factors.^

Extinction times for a general birth, death and catastrophe process

The birth, death and catastrophe process is an extension of the birth-death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

Approximating persistence in a general class of population processes

We provide a general framework for estimating persistence in populations which may be affected by catastrophic events, and which are either unbounded or have very large ceilings. We model the population using a birth-death process modified to allow for downward jumps of arbitrary size. For such processes, it is typically necessary to truncate the process in order to make the evaluation of expected extinction times (and higher-order moments) computationally feasible. Hence, we give particular attention to the selection of a cut-off point at which to truncate the process, and we present a simple method for obtaining quantitative indicators of the suitability of a chosen cut-off. (c) 2005 Elsevier Inc. All rights reserved.

Survival time of random walk in random environment among soft obstacles

We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d. random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the survival time in the general d-dimensional case. We then consider a simplified one-dimensional model (where transition probabilities and obstacles are independent and the RWRE only moves to neighbour sites), and obtain finer results for the tail of the survival time. In addition, we study also the ""mixed"" probability measures (quenched with respect to the obstacles and annealed with respect to the transition probabilities and vice-versa) and give results for tails of the survival time with respect to these probability measures. Further, we apply the same methods to obtain bounds for the tails of hitting times of Branching Random Walks in Random Environment (BRWRE).

Control of interceptive actions is based on expectancy of time to target arrival

In this study the hypothesis that interceptive movements are controlled on the basis of expectancy of time to target arrival was tested. The study was conducted through assessment of temporal errors and kinematics of interceptive movements to a moving virtual target. Initial target velocity was kept unchanged in part of the trials, and in the others it was decreased 300 ms before the due time of target arrival at the interception position, increasing in 100 ms time to target arrival. Different probabilities of velocity decrease ranging from 25 to 100% were compared. The results revealed that while there were increasing errors between probabilities of 25 and 75% for unchanged target velocity, the opposite relationship was observed for target velocity decrease. Kinematic analysis indicated that movement timing adjustments to target velocity decrease were made online. These results support the conception that visuomotor integration in the interception of moving targets is mediated by an internal forward model whose weights can be flexibly adjusted according to expectancy of time to target arrival.

Constraints on the spatiotemporal accuracy of interceptive action: effects of target size on hitting a moving object

Results of two experiments are reported that examined how people respond to rectangular targets of different sizes in simple hitting tasks. If a target moves in a straight line and a person is constrained to move along a linear track oriented perpendicular to the targetrsquos motion, then the length of the target along its direction of motion constrains the temporal accuracy and precision required to make the interception. The dimensions of the target perpendicular to its direction of motion place no constraints on performance in such a task. In contrast, if the person is not constrained to move along a straight track, the targetrsquos dimensions may constrain the spatial as well as the temporal accuracy and precision. The experiments reported here examined how people responded to targets of different vertical extent (height): the task was to strike targets that moved along a straight, horizontal path. In experiment 1 participants were constrained to move along a horizontal linear track to strike targets and so target height did not constrain performance. Target height, length and speed were co-varied. Movement time (MT) was unaffected by target height but was systematically affected by length (briefer movements to smaller targets) and speed (briefer movements to faster targets). Peak movement speed (Vmax) was influenced by all three independent variables: participants struck shorter, narrower and faster targets harder. In experiment 2, participants were constrained to move in a vertical plane normal to the targetrsquos direction of motion. In this task target height constrains the spatial accuracy required to contact the target. Three groups of eight participants struck targets of different height but of constant length and speed, hence constant temporal accuracy demand (different for each group, one group struck stationary targets = no temporal accuracy demand). On average, participants showed little or no systematic response to changes in spatial accuracy demand on any dependent measure (MT, Vmax, spatial variable error). The results are interpreted in relation to previous results on movements aimed at stationary targets in the absence of visual feedback.

Hitting a moving target: Perception and action in the timing of rapid interceptions

Different interceptive tasks and modes of interception (hitting or capturing) do not necessarily involve similar control processes. Control based on preprogramming of movement parameters is possible for actions with brief movement times but is now widely rejected; continuous perceptuomotor control models are preferred for all types of interception. The rejection of preprogrammed control and acceptance of continuous control is evaluated for the timing of rapidly executed, manual hitting actions. It is shown that a preprogrammed control model is capable of providing a convincing account of observed behavior patterns that avoids many of the arguments that have been raised against it. Prominent continuous perceptual control models are analyzed within a common framework and are shown to be interpretable as feedback control strategies. Although these models can explain observations of on-line adjustments to movement, they offer only post hoc explanations for observed behavior patterns in hitting tasks and are not directly supported by data. It is proposed that rapid manual hitting tasks make up a class of interceptions for which a preprogrammed strategy is adopted-a strategy that minimizes the role of visual feedback. Such a strategy is effective when the task demands a high degree of temporal accuracy.

Path integrals for continuous-time Markov chains

This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.

Integrals for continuous-time Markov chains

This paper presents a method of evaluating the expected value of a path integral for a general Markov chain on a countable state space. We illustrate the method with reference to several models, including birth-death processes and the birth, death and catastrophe process. (C) 2002 Elsevier Science Inc. All rights reserved.

Criteria for hitting probabilities with applications to systems of stochastic wave equations

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension k >- 1 driven by a d-dimensional spatially homogeneous additive Gaussian noise that is white in time and colored in space.

The accuracy of interceptive action in time and space

Hitting a moving target demands that movement is both spatially and temporally accurate. Recent experiments have begun to reveal how performance of such actions depends on the spatial and temporal accuracy requirements of the task. The results suggest a simple strategy for achieving spatiotemporal accuracy using brief, high-speed movements.