B. Gail Ivanoff

What is a multi-parameter renewal process?

The concept of the renewal property is extended to processes indexed by a multidimensional time parameter. The definition given includes not only partial sum processes, but also Poisson processes and many other point processes whose jump points are not totally ordered. A new version of the waiting time paradox is proven for multidimensional Poisson processes, and is shown to imply the renewal property. Finally, martingale properties of renewal processes are studied.

The function space D([O, ∞)ρ, E)

The Skorokhod topology is extended to the function space D([0, ∞)ρ, E) of functions, from [0, ∞)ρ to a complete separable metric space E, which are “continuous from above with limits from below. Criteria for tightness are developed. The case in which E is a product space is considered, and conditions under which tightness may be proven componentwise are given. Various applications are studied, including a multidimensional version of Donskers Theorem, and a functional Central Limit Theorem for a multitype Poisson cluster process.

Stopping and set-indexed local martingales

Set-indexed local martingales are defined and studied. We present some optional sampling theorems for strong martingales, martingales and weak martingales. The class of set-indexed processes which are locally of class (D) is introduced. A Doob-Meyer decomposition is obtained: any local weak submartingale has a unique decomposition into the sum of a local weak martingale and a local predictable increasing process. Finally some examples are given.

A martingale characterization of the set-indexed poisson process

A martingale characterization of the set-indexed Poisson process is proved: a set-indexed point process is a Poisson process if and only if there exists a deterministic and increasing process such that the difference is a strong martingale

Doob-meyer decomposition for set-indexed submartingales

Set-indexed martingales and submartingales are defined and studied. The admissible function of a submartingale is defined and some class (D) conditions are given which allow the extension of the function to a σ-additive measure on the predictable σ-algebra. Then, we prove a Doob-Meyer decomposition: A set-indexed submartingale can be decomposed into the sum of a weak martingale and an increasing process. A hypothesis of predictability ensures the uniqueness of this decomposition. An explicit construction of the increasing process associated with a submartingale is given. Finally, some remarks, about quasimartingales are discussed.

Predictability and stopping on lattices of sets

SummaryAs a first step in the development of a general theory of set-indexed martingales, we define predictability on a general space with respect to a filtration indexed by a lattice of sets. We prove a characterization of the predictable σ-algebra in terms of adapted and “left-continuous” processes without any form of topology for the index set. We then define a stopping set and show that it is a natural generalization of the stopping time; in particular, the predictable σ-algebra can be characterized by various stochastic intervals generated by stopping sets.

Stopping times and tightness for multiparameter martingales

Two major results on stopping times and tightness of Aldous are generalized to strong martingales with a multidimensional time parameter

Optimal detection of a change-set in a spatial Poisson process

We generalize the classic change-point problem to a “change-set” framework: a spatial Poisson process changes its intensity on anunobservable random set. Optimal detection of the set is defined bymaximizing the expected value of a gain function. In the case that theunknown change-set is defined by a locally finite set of incomparablepoints, we present a sufficient condition for optimal detection of theset using multiparameter martingale techniques. Two examples arediscussed.

Poisson convergence for point processes on the plane

A compensator is defined for a point process in two dimensions. It is shown that a Poisson process is characterized by a continuous deterministic compensator. Sufficient conditions are given for convergence in distribution of a sequence of two-dimensional point processes in the Skorokhod topology to a Poisson process when the corresponding sequence of compensators converges pointwise in probability to a continuous deterministic function.

The multitype branching diffusion

The multitype branching diffusion (MBD) is considered. A review of the general theory of multitype point processes is given in Section 2, and spatial central limit theorems for homogeneous infinitely divisible processes are proven in Section 3. In Section 4, the MBD is defined, and equations for its first four factorial moment density functions are found. The behaviour of the mean and covariance functionals as time approaches infinity is studied. The MBD with immigration (MBDI) is introduced in Section 5. The existence of a steady state is proven, and spatial central limit theorems are developed for the MBDI.