Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Volume I first provides a compact survey on classical stochastic geometry models, with a main focus on spatial shot-noise processes, coverage processes and random tessellations. It then focuses on signal to interference noise ratio (SINR) stochastic geometry, which is the basis for the modeling of wireless network protocols and architectures considered in Volume II. It also contains an appendix on mathematical tools used throughout Stochastic Geometry and Wireless Networks, Volumes I and II.

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Stochastic Geometry and Wireless Networks: Volume I Theory

We discuss briefly the very interesting concept of Brownian distance covariance developed by Szekely and Rizzo (2009) and describe two possible extensions. The first extension is for high dimensional data that can be coerced into a Hilbert space, including certain high throughput screening and functional data settings. The second extension involves very simple modifications that may yield increased power in some settings. We commend Szekely and Rizzo for their very interesting work and recognize that this general idea has potential to have a large impact on the way in which statisticians evaluate dependency in data.

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Brownian distance covariance

Energy statistics: A class of statistics based on distances

The Poisson process, a core object in modern probability, enjoys a richer theory than is sometimes appreciated. This volume develops the theory in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the Poisson process. Also discussed are applications and related topics in stochastic geometry, including stationary point processes, the Boolean model, the Gilbert graph, stable allocations, and hyperplane processes. Comprehensive, rigorous, and self-contained, this text is ideal for graduate courses or for self-study, with a substantial number of exercises for each chapter. Mathematical prerequisites, mainly a sound knowledge of measure-theoretic probability, are kept in the background, but are reviewed comprehensively in the appendix. The authors are well-known researchers in probability theory; especially stochastic geometry. Their approach is informed both by their research and by their extensive experience in teaching at undergraduate and graduate levels.

Lectures on the Poisson Process

Introduction General Theory Generalities. Predictability. Martingales. Decompositions and Quadratic Variation Martingale Characterizations. Generalizations of Martingales Weak Convergence. Weak Convergence of Set-Indexed Processes Limit Theorems for Point Processes Martingale Central Limit Theorems References Index.

Set-Indexed Martingales

Watanabe proved that if $X_t$ is a point process such that $X_t - t$ is a martingale, then $X_t$ is a Poisson process and this result was generalized by Bremaud for doubly stochastic Poisson processes. Here we define two-parameter point processes and extend this property without needing the strong martingale condition. Using this characterization, we study the problem of transforming a two-parameter point process into a two-parameter Poisson process by means of a family of stopping lines as a time change. Nualart and Sanz gave conditions in order to transform a square integrable strong martingale into a Wiener process. Here, we do the same for the Poisson process by a similar method but under more general conditions.

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A Characterization of the Spatial Poisson Process and Changing Time

We define and prove the existence of a fractional Brownian motion in- dexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Levy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove sta- tionarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no "really nice" set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.

http://u.cs.biu.ac.il/~merzbach/Publications/56fractionalbrownian.pdf

A Set-indexed Fractional Brownian Motion

We present new properties for the Fractional Poisson process and the Fractional Poisson field on the plane. A martingale characterization for Fractional Poisson processes is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane.

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Fractional Poisson fields and Martingales

We present new properties for the Fractional Poisson process and the Fractional
Poisson field on the plane. A martingale characterization for Fractional Poisson
processes is given. We extend this result to Fractional Poisson fields, obtaining some
other characterizations. The fractional differential equations are studied. We consider
a more general Mixed-Fractional Poisson process and show that this process is the
stochastic solution of a system of fractional differential-difference equations. Finally,
we give some simulations of the Fractional Poisson field on the plane.

/pdf/fractional-poisson-fields-and-martingales-2if1gjhnu9.pdf

Ely Merzbach

Papers

Set-Indexed Martingales

A Characterization of the Spatial Poisson Process and Changing Time

A Set-indexed Fractional Brownian Motion

Fractional Poisson fields and Martingales

Fractional Poisson Fields and Martingales