Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Stochastic differential equations and diffusion processes

An Introduction to Probability Theory and its Applications, Volume I

A first course in stochastic processes

We consider a branching model introduced by M. Kimmel for cell division with parasite infection. Cells contain proliferating parasites which are shared randomly between the two daughter cells when they divide. We determine the probability that the organism recovers, meaning that the asymptotic proprotion of contaminated cells vanishes. We study the tree of contaminated cells, give the asymptotic number of contaminated cells and the asymptotic proportions of contaminated cells with a given number of parasites. This depends on domains inherited from the behavior of branching processes in random environment (BPRE) and given by the bivariate value of the means of parasite offsprings. In one of these domains, the convergence of proportions holds in probability, the limit is deterministic and given by the Yaglom quasistationary distribution. Moreover we get an interpretation of the limit of the Q-process as the size-biased quasistationary distribution.

/pdf/proliferating-parasites-in-dividing-cells-kimmel-s-branching-3ass7ba049.pdf

Proliferating parasites in dividing cells : Kimmel's branching model revisited.

We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton--Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, associated with branching events of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time t favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Levy processes.

Limit theorems for Markov processes indexed by continuous time Galton--Watson trees

We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time $t$. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. This latter has the same generator as the Markov process along the branches plus additional branching events, associated with jumps of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time $t$ favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Levy processes and ancestral lineages.

/pdf/limit-theorems-for-markov-processes-indexed-by-continuous-2rg60914bq.pdf

Limit theorems for Markov processes indexed by continuous time Galton–Watson trees

We now deal with a continuous time model for dividing cells which are infected by parasites. We assume that parasites proliferate in the cells and that their lifetimes are much shorter than the cell lifetimes. The quantity of parasites (X t : t ≥ 0) in a cell is modeled by a Feller diffusion (see Chapter 3 and Definition 4.1). The cells divide in continuous time at rate τ(x) which may depend on the quantity of parasites x that they contain. When a cell divides, a random fraction F of the parasites goes in the first daughter cell and a fraction (1 − F) in the second one. More generally, splitting Feller diffusion may model the quantity of some biological content which grows (without resource limitation) in the cells and is shared randomly when the cells divide (for example, proteins, nutriments, energy or extrachromosomal rDNA circles in yeast).

/pdf/branching-feller-diffusion-for-cell-division-with-parasite-4sxx0rs2s5.pdf

Branching Feller diffusion for cell division with parasite infection

We consider continuous state branching processes (CSBP's) with additional multiplicative jumps modeling dramatic  events in a random environment. These jumps  are described by a Levy process with bounded variation paths. We construct the associated class of processes as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and make appear new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish when it becomes extinct. For a class of processes for which extinction and absorption coincide (including the $\alpha$ stable CSBP's plus a drift), we determine the speed of extinction. Four regimes appear, as in the case of branching processes in random environment in discrete time and space.The proofs rely on a fine study of the asymptotic behavior of exponential functionals of Levy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection.

/pdf/on-the-extinction-of-continuous-state-branching-processes-1esnm1b33j.pdf

Vincent Bansaye

Papers

Proliferating parasites in dividing cells : Kimmel's branching model revisited.

Limit theorems for Markov processes indexed by continuous time Galton--Watson trees

Limit theorems for Markov processes indexed by continuous time Galton–Watson trees

Branching Feller diffusion for cell division with parasite infection

On the extinction of Continuous State Branching Processes with catastrophes