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334 results on '"Waymire, Edward C."'

1. Transformation of Stochastic Recursions and Critical Phenomena in the Analysis of a Class of Mean Flow Equations

Author

Dascaliuc, Radu, Pham, Tuan N., Thomann, Enrique, and Waymire, Edward C.

Subjects
Mathematics - Probability, Mathematics - Analysis of PDEs, 60J80 60J85 60H30 60J85 35Q30 92D25
Abstract

This paper investigates, via methods from the theory of probability on trees, critical phenomena in stochastic cascade models of Yule type, and applies these methods to the problem of uniqueness and nonuniqueness of solutions of particular mean flow equations. For linear equations, Feller's classic discovery of the relationship between critical stochastic explosion phenomena for Markov processes and uniqueness and nonuniqueness of solutions to the associated Kolmogorov equation partly illustrates the spirit of the present paper. New methods are introduced to explore the effects that stochastic critical phenomena, such as stochastic explosion, hyperexplosion, t-leaf percolation, have on their corresponding mean flow equations. These methods are illustrated in the context of two non-local differential equations that are related by linearization, the pantograph and the $\alpha$-Riccati equations. In particular, stochastic recursions, stochastic Picard iterations and transformation of solutions processes associated with these equations, are used to establish a `one-to-many principle' for their solutions. The development also includes an application of Feller's classical theory of jump Markov processes to a certain class of pantograph equations that appears to be new., Comment: 44 pages, 9 figures

Published
2024

2. Errata to Stochastic explosion and non-uniqueness for $\alpha$-Riccati equation

Author

Dascaliuc, Radu, Pham, Tuan N., Thomann, Enrique A., and Waymire, Edward C.

Subjects
Mathematics - Probability, 60H30, 34K12, 60J80
Abstract

An error occurs in a part of the statement and proof of Proposition 2.2 in Jour. Math. Anal. and Appl., 476, (2019), 53-85 that is corrected in this erratum. The revised result reveals a new and unexpected critical phenomenon, having further implications for non-uniqueness of solutions to a nonlinear differential equation of the Riccati type.

Published
2023
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3. Doubly stochastic Yule cascades (Part II): The explosion problem in the non-reversible case

Author

Dascaliuc, Radu, Pham, Tuan N., Thomann, Enrique, and Waymire, Edward C.

Subjects
Mathematics - Probability, 60H30, 60J80
Abstract

We analyze the explosion problem for a class of stochastic models introduced in Part I (arXiv:2103.06912), referred to as doubly stochastic Yule cascades. These models arise naturally in the construction of solutions to evolutionary PDEs as well as in purely probabilistic first passage percolation phenomena having a Markov-type statistical dependence, new for this context. Using cut-set arguments and a greedy algorithm, we respectively establish criteria for non-explosion and explosion without requiring the time-reversibility of the underlying branching Markov chain (a condition required in Part I). Notable applications include the explosion of the self-similar cascade of the Navier-Stokes equations in dimension $d=3$ and non-explosion in dimensions $d\ge 12$., Comment: 34 pages

Published
2021

4. Doubly Stochastic Yule Cascades (Part I): The explosion problem in the time-reversible case

Author

Dascaliuc, Radu, Pham, Tuan N., Thomann, Enrique, and Waymire, Edward C.

Subjects
Mathematics - Probability, 60J80 60J85 60H30 60J85 35Q30 92D25
Abstract

Motivated by the probabilistic methods for nonlinear differential equations introduced by McKean (1975) for the Kolmogorov-Petrovski-Piskunov (KPP) equation, and by Le Jan and Sznitman (1997) for the incompressible Navier-Stokes equations, we identify a new class of stochastic cascade models, referred to as Doubly Stochastic Yule cascades. We establish non-explosion criteria under the assumption that the randomization of Yule intensities from generation to generation is by an ergodic time-reversible Markov process. In addition to the cascade models that arise in the analysis of certain deterministic nonlinear differential equations, this model includes the multiplicative branching random walks, the branching Markov processes, and the stochastic generalizations of the percolation and/or cell aging models introduced by Aldous and Shields (1988) and independently by Athreya (1985)., Comment: 23 pages

Published
2021
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7. Continuous Parameter Martingales

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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8. Optional Stopping of (Sub)Martingales

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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9. Coupling Methods for Markov Chains and the Renewal Theorem for Lattice Distributions

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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10. Bienaymé–Galton–Watson Simple Branching Process and Extinction

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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11. The Kolmogorov–Chentsov Theorem and Sample Path Regularity

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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12. Random Walk, Brownian Motion, and the Strong Markov Property

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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13. The Poisson Process, Compound Poisson Process, and Poisson Random Field

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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14. Multidimensional Random Walk

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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15. Special Topic: Incompressible Navier–Stokes Equations and the Le Jan–Sznitman Cascade

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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16. The Simple Random Walk II: First Passage Times

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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17. Special Topic: Fractional Brownian Motion and/or Trends: The Hurst Effect

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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18. Special Topic: Ruin Problems in Insurance

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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19. The Simple Random Walk I: Associated Boundary Value Distributions, Transience, and Recurrence

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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20. Special Topic: The Geometric Random Walk and the Binomial Tree Model of Mathematical Finance

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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21. Special Topic: A Comprehensive Renewal Theory for General Random Walks

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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22. Special Topic: Optimal Stopping Rules

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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23. Special Topic: Branching Random Walk, Polymers, and Multiplicative Cascades

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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24. Brownian Motion on the Half-Line: Absorption and Reflection

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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25. The Brownian Bridge

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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26. Special Topic: Bienaymé–Galton–Watson Simple Branching Process and Excursions

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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27. First Passage Time Distributions for Brownian Motion with Drift and a Local Limit Theorem

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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28. ArcSine Law Asymptotics

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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29. What Is a Stochastic Process?

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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30. Stochastic Calculus for Point Processes and a Martingale Characterization of the Poisson Process

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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31. Growth of Supercritical Bienaymé–Galton–Watson Simple Branching Processes

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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32. The Functional Central Limit Theorem (FCLT)

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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33. The Upcrossings Inequality and (Sub)Martingale Convergence

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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34. Martingales: Definitions and Examples

Author

Bhattacharya, Rabi, Waymire, Edward C., Vakil, Ravi, Series Editor, Adem, Alejandro, Advisory Editor, Eisenbud, David, Advisory Editor, Hall, Brian C., Advisory Editor, Hersh, Patricia, Advisory Editor, Lagarias, Jeffrey C., Advisory Editor, Malinnikova, Eugenia, Advisory Editor, Ono, Ken, Advisory Editor, Quastel, Jeremy, Advisory Editor, Simon, Barry, Advisory Editor, Weintraub, Steven H., Advisory Editor, Matchett Wood, Melanie, Advisory Editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2021
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35. When Fourth Moments Are Enough

Author

Jennings-Shaffer, Chris, Skinner, Dane R., and Waymire, Edward C.

Subjects
Mathematics - Probability, Mathematics - Statistics Theory
Abstract

This note concerns a somewhat innocent question motivated by an observation concerning the use of Chebyshev bounds on sample estimates of $p$ in the binomial distribution with parameters $n,p$. Namely, what moment order produces the best Chebyshev estimate of $p$? If $S_n(p)$ has a binomial distribution with parameters $n,p$, there it is readily observed that ${\rm argmax}_{0\le p\le 1}{\mathbb E}S_n^2(p) = {\rm argmax}_{0\le p\le 1}np(1-p) = \frac12,$ and ${\mathbb E}S_n^2(\frac12) = \frac{n}{4}$. Rabi Bhattacharya observed that while the second moment Chebyshev sample size for a $95\%$ confidence estimate within $\pm 5$ percentage points is $n = 2000$, the fourth moment yields the substantially reduced polling requirement of $n = 775$. Why stop at fourth moment? Is the argmax achieved at $p = \frac12$ for higher order moments and, if so, does it help, and compute $\mathbb{E}S_n^{2m}(\frac12)$? As captured by the title of this note, answers to these questions lead to a simple rule of thumb for best choice of moments in terms of an effective sample size for Chebyshev concentration inequalities.

Published
2017

36. A Delayed Yule Process

Author

Dascaliuc, Radu, Michalowski, Nicholas, Thomann, Enrique, and Waymire, Edward C.

Subjects
Mathematics - Probability, 60J80 60G44
Abstract

In now classic work, David Kendall (1966) recognized that the Yule process and Poisson process could be related by a (random) time change. Furthermore, he showed that the Yule population size rescaled by its mean has an almost sure exponentially distributed limit as $t\to \infty$. In this note we introduce a class of coupled delayed Yule processes parameterized by $0 < \alpha \le 1$ that includes the Poisson process at $\alpha = {1/2}$. Moreover we extend Kendall's limit theorem to include a larger class of positive martingales derived from functionals that gauge the population genealogy. A somewhat surprising connection with the Holley-Liggett smoothing transformation also emerges in this context. Specifically, the latter is exploited to uniquely characterize the moment generating functions of distributions of the limit martingales, generalizing Kendall's mean one exponential limit.

Published
2016

38. Continuity of Local Time: An applied perspective

Author

Ramirez, Jorge M., Waymire, Edward C., and Thomann, Enrique A.

Subjects
Mathematics - Probability, 60G17, 60G51
Abstract

Continuity of local time for Brownian motion ranks among the most notable mathematical results in the theory of stochastic processes. This article addresses its implications from the point of view of applications. In particular an extension of previous results on an explicit role of continuity of (natural) local time is obtained for applications to recent classes of problems in physics, biology and finance involving discontinuities in a dispersion coefficient. The main theorem and its corollary provide physical principles that relate macro scale continuity of deterministic quantities to micro scale continuity of the (stochastic) local time., Comment: To appear in: "The fascination of Probability, Statistics and Their Applications. In honour of Ole E. Barndorff-Nielsen on his 80th birthday"

Published
2015

39. Symmetry Breaking and Uniqueness for the Incompressible Navier-Stokes Equations

Author

Dascaliuc, Radu, Michalowski, Nicholas, Thomann, Enrique, and Waymire, Edward C.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Probability, Physics - Fluid Dynamics, 35Q30, 76D03, 60J80
Abstract

The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the uniqueness problem for Navier-Stokes equations. In particular, new branching Markov chains, including a dilogarithmic branching random walk on the multiplicative group $(0,\infty)$, naturally arise as a result of this investigation., Comment: 30 pages, 2 figures; to appear in Chaos

Published
2015
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41. Advection-Dispersion Across Interfaces

Author

Ramirez, Jorge M., Thomann, Enrique A., and Waymire, Edward C.

Subjects
Mathematics - Statistics Theory, Statistics - Methodology
Abstract

This article concerns a systemic manifestation of small scale interfacial heterogeneities in large scale quantities of interest to a variety of diverse applications spanning the earth, biological and ecological sciences. Beginning with formulations in terms of partial differential equations governing the conservative, advective-dispersive transport of mass concentrations in divergence form, the specific interfacial heterogeneities are introduced in terms of (spatial) discontinuities in the diffusion coefficient across a lower-dimensional hypersurface. A pathway to an equivalent stochastic formulation is then developed with special attention to the interfacial effects in various functionals such as first passage times, occupation times and local times. That an appreciable theory is achievable within a framework of applications involving one-dimensional models having piecewise constant coefficients greatly facilitates our goal of a gentle introduction to some rather dramatic mathematical consequences of interfacial effects that can be used to predict structure and to inform modeling., Comment: Published in at http://dx.doi.org/10.1214/13-STS442 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2013
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42. On The Drift Paradox in a Regime-Switching Model

Author

Felder, William and Waymire, Edward C.

Subjects
Mathematics - Probability
Abstract

This note is motivated by the article by F. Lutscher, E. Pachepsky, and M. Lewis (2005), The Effect of Dispersal Patterns on Stream Populations SIAM Rev. Vol. 47 No. 4 pp. 749-772 on the drift paradox. We consider the case of a regime switching probabilistic model for a population of organisms living in a one dimensional environment with drift towards an absorbing boundary of the type introduced by Lutscher et al. (2005). In particular, the two regimes consist of birth/death style demographics governing the evolution of the immobile regime, and some form of advection-dispersion governing the evolution of the mobile regime, together with a regime-switching mechanism linking the two. In the present note it is shown for the regime-switching model, and in contrast to the results in the afore cited work, for any finite advection speed, no matter how large, there is a finite critical length of the domain above which the population can persist.

Published
2013

45. First Passage Times and Breakthrough Curves Associated with Interfacial Phenomena

Author

Waymire, Edward C., Appuhamillage, Thilanka A., Bokil, Vrushali A., Thomann, Enrique A., and Wood, Brian D.

Subjects
Mathematics - Probability, Condensed Matter - Statistical Mechanics
Abstract

Advection and dispersion in highly heterogeneous environments involving interfacial discontinuities in the corresponding drift and dispersion rates are described through disparate examples from the physical and biological sciences. A mathematical framework is formulated to address specific empirical phenomena involving first passage time and occupation time functionals observed in relation to the interfacial parameters.

Published
2011

46. Tree polymers in the infinite volume limit at critical strong disorder

Author

Johnson, Torrey and Waymire, Edward C

Subjects
Mathematics - Probability
Abstract

The a.s. existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walk. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the a.s. asymptotic variance of the polymer path under strong disorder are also provided., Comment: Final version

Published
2010
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49. Fourier Series, Fourier Transform, and Characteristic Functions

Author

Bhattacharya, Rabi, Waymire, Edward C., Axler, Sheldon, Series editor, Casacuberta, Carles, Series editor, MacIntyre, Angus, Series editor, Ribet, Kenneth, Series editor, Sabbah, Claude, Series editor, Süli, Endre, Series editor, Woyczyński, Wojbor A., Series editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2016
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50. Classical Zero–One Laws, Laws of Large Numbers and Large Deviations

Author

Bhattacharya, Rabi, Waymire, Edward C., Axler, Sheldon, Series editor, Casacuberta, Carles, Series editor, MacIntyre, Angus, Series editor, Ribet, Kenneth, Series editor, Sabbah, Claude, Series editor, Süli, Endre, Series editor, Woyczyński, Wojbor A., Series editor, Bhattacharya, Rabi, and Waymire, Edward C.

Published
2016
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