ON THE CONVERGENCE OF DISCRETE PROCESSES WITH MULTIPLE INDEPENDENT VARIABLES

We discuss discrete stochastic processes with two independent variables: one is the standard symmetric random walk, and the other is the Poisson process. Convergence of discrete stochastic processes is analysed, such that the symmetric random walk tends to the standard Brownian motion. We show that a discrete analogue of Ito’s formula converges to the corresponding continuous formula.

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Stochastic Processes with a Particular Type of Variograms

Research Letters in Signal Processing ◽

10.1155/2007/61579 ◽

2007 ◽

Vol 2007 ◽

pp. 1-5 ◽

Cited By ~ 1

Author(s):

Chunsheng Ma

Keyword(s):

Stochastic Process ◽

Brownian Motion ◽

Stochastic Processes ◽

Series Expansion ◽

Real Line ◽

Random Fields ◽

The Other ◽

Simple Construction ◽

The Real ◽

Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

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Introductory remarks

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

10.1098/rsta.1990.0123 ◽

1990 ◽

Vol 332 (1627) ◽

pp. 417-418

Keyword(s):

Probability Theory ◽

Brownian Motion ◽

Random Walk ◽

Stochastic Processes ◽

Statistical Physics ◽

Late 19Th Century ◽

Fields Of Study ◽

Wide Range ◽

Influential Paper ◽

Telephone Traffic

Stochastic processes are systems that evolve in time probabilistically; their study is the ‘dynamics’ of probability theory as contrasted with rather more traditional ‘static’ problems. The analysis of stochastic processes has as one of its main origins late 19th century statistical physics leading in particular to studies of random walk and brownian motion (Rayleigh 1880; Einstein 1906) and via them to the very influential paper of Chandrasekhar (1943). Other strands emerge from the work of Erlang (1909) on congestion in telephone traffic and from the investigations of the early mathematical epidemiologists and actuarial scientists. There is by now a massive general theory and a wide range of special processes arising from applications in many fields of study, including those mentioned above. A relatively small part of the above work concerns techniques for the analysis of empirical data arising from such systems.

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Problèmes de premier passage résolubles par la méthode de séparation des variables

Journal of Applied Probability ◽

10.1017/s0021900200102827 ◽

1995 ◽

Vol 32 (02) ◽

pp. 337-348

Author(s):

Mario Lefebvre

Keyword(s):

Brownian Motion ◽

Stochastic Processes ◽

Generating Function ◽

Moment Generating Function ◽

First Passage Times ◽

Standard Brownian Motion ◽

First Passage ◽

The Moment ◽

Passage Times

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2 dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

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An Investigation of the Effect of Latency on the Operator’s Trust and Performance for Manual Multi-robot Teleoperated Tasks

Proceedings of the Human Factors and Ergonomics Society Annual Meeting ◽

10.1177/1541931213601579 ◽

2017 ◽

Vol 61 (1) ◽

pp. 390-394 ◽

Cited By ~ 1

Author(s):

Hunter Rogers ◽

Amro Khasawneh ◽

Jeffery Bertrand ◽

Kapil Chalil Madathil

Keyword(s):

Confidence Level ◽

Search And Rescue ◽

The Other ◽

Repeated Measure ◽

Independent Variables ◽

Subjective Workload ◽

Dependent Variables ◽

Independent Variable ◽

And Performance ◽

Two Independent Variables

Latency is an important factor when conducting teleoperated missions. This study investigates the effects of latency on a set of dependent variables: performance (measured by time and number of errors), subjective workload, trust, and usability. These measures were tested in a simulated search-and-rescue mission over two levels of two independent variables. One independent variable was the number of robots – one or two (within-subject), and the other independent variable was latency – simulations with and without latency (between-subject.) The significant effect of the independent variables on the dependent variables were checked using repeated measure two-way ANOVA with a confidence level of 95%. The data determined any significant effects that latency and/or the number of robots had on such factors as errors, dependability, reliability, harmful outcomes, temporal demand, and frustration.

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General approach to the sensitivity of the optics of an eye to change in elementary parameters with application to the Gaussian optics of a reduced eye

African Vision and Eye Health ◽

10.4102/aveh.v68i4.169 ◽

2009 ◽

Vol 68 (4) ◽

Author(s):

W. F. Harris

Keyword(s):

Optical Properties ◽

Index Of Refraction ◽

The Other ◽

Radius Of Curvature ◽

Corneal Curvature ◽

Independent Variables ◽

Gaussian Optics ◽

Clear Statement ◽

Two Independent Variables ◽

General Method

Optical properties of the eye, including the refractive compensation, for example, may change if there are changes in any of the components that make up the eye. The sensitivity to such changes isquantified via the derivative. This paper employs the reduced eye and Gaussian optics to illustrate a general method for the analysis of sensitivity in eyes. The method requires a clear statement of the dependent variable as a function of independent variables. A symbolism is offered that makes the function unambiguous. Sensitivities are determined for the fundamental optical properties,the transference and the corneal-plane refractivecompensation of a reduced eye to change in corneal power, curvature and radius of curvature and to change in axial length and index of refraction. Emsley’s reduced eye is examined in particular. Its corneal-plane refractive compensation has a sensitivity of 135 −135 D to change in refractive index, 1 −2.7 D/mm to change in length and 135 −1/3 to change in corneal curvature when the other two independent variables are held fixed. The method has the potential to develop guidelines that are useful clini-cally.

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Lengths and heights of random walk excursions

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3337 ◽

2003 ◽

Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽

Author(s):

Endre Csáki ◽

Yueyun Hu

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Limiting Distributions ◽

Symmetric Random Walk ◽

International Audience

International audience Consider a simple symmetric random walk on the line. The parts of the random walk between consecutive returns to the origin are called excursions. The heights and lengths of these excursions can be arranged in decreasing order. In this paper we give the exact and limiting distributions of these ranked quantities. These results are analogues of the corresponding results of Pitman and Yor [1997, 1998, 2001] for Brownian motion.

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Limiting stochastic processes of shift-periodic dynamical systems

Royal Society Open Science ◽

10.1098/rsos.191423 ◽

2019 ◽

Vol 6 (11) ◽

pp. 191423

Author(s):

Julia Stadlmann ◽

Radek Erban

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Dynamical Systems ◽

Stochastic Processes ◽

Discrete Time ◽

Real Line ◽

Continuous Time ◽

Initial Distribution ◽

Dynamical Behaviour ◽

One Dimensional

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences x n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ give a discrete-time random walk for a suitable initial distribution of x 0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.

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Bounds of Double Integral Dynamic Inequalities in Two Independent Variables on Time Scales

Discrete Dynamics in Nature and Society ◽

10.1155/2011/732164 ◽

2011 ◽

Vol 2011 ◽

pp. 1-21 ◽

Cited By ~ 1

Author(s):

S. H. Saker

Keyword(s):

Time Scales ◽

Unknown Function ◽

The Other ◽

Dynamic Equations ◽

Double Integral ◽

Independent Variables ◽

Explicit Bounds ◽

Quantitative Properties ◽

The One ◽

Two Independent Variables

Our aim in this paper is to establish some explicit bounds of the unknown function in a certain class of nonlinear dynamic inequalities in two independent variables on time scales which are unbounded above. These on the one hand generalize and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of partial dynamic equations on time scales. Some examples are considered to demonstrate the applications of the results.

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Problèmes de premier passage résolubles par la méthode de séparation des variables

Journal of Applied Probability ◽

10.2307/3215292 ◽

1995 ◽

Vol 32 (2) ◽

pp. 337-348 ◽

Cited By ~ 1

Author(s):

Mario Lefebvre

Keyword(s):

Brownian Motion ◽

Stochastic Processes ◽

Generating Function ◽

Moment Generating Function ◽

First Passage Times ◽

Standard Brownian Motion ◽

First Passage ◽

The Moment ◽

Passage Times

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

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On a new Ostrowski type inequality in two independent variables

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.32.2001.367 ◽

2001 ◽

Vol 32 (1) ◽

pp. 45-49

Author(s):

B. G. Pachpatte

Keyword(s):

Integral Inequality ◽

Type Inequality ◽

Discrete Analogue ◽

Independent Variables ◽

Ostrowski Type Inequality ◽

Two Independent Variables

In this note a new integral inequality of Ostrowski type in two independent variables is established. The discrete analogue of the main result is also given.

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