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

The proteasome is an essential part of our immune surveillance mechanisms: by generating peptides from intracellular antigens it provides peptides that are then 'presented' to T cells. But proteasomes--the waste-disposal units of the cell--typically do not generate peptides for antigen presentation with high efficiency. How, then, does the proteasome adapt to serve the immune system well?

Antigen processing by the proteasome

Diffusion processes and their sample paths

Mean field games are limit models for symmetric $N$-player games with interaction of mean field type as $N\to \infty $. The limit relation is often understood in the sense that a solution of a mean field game allows to construct approximate Nash equilibria for the corresponding $N$-player games. The opposite direction is of interest, too: When do sequences of Nash equilibria converge to solutions of an associated mean field game? In this direction, rigorous results are mostly available for stationary problems with ergodic costs. Here, we identify limit points of sequences of certain approximate Nash equilibria as solutions to mean field games for problems with Ito-type dynamics and costs over a finite time horizon. Limits are studied through weak convergence of associated normalized occupation measures and identified using a probabilistic notion of solution for mean field games.

/pdf/on-the-connection-between-symmetric-n-player-games-and-mean-m7oyphm3s4.pdf

On the connection between symmetric $N$-player games and mean field games

We study large deviation properties of systems of weakly interacting particles modeled by Ito stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean–Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay.

/pdf/large-deviation-properties-of-weakly-interacting-processes-4xeyqm4uf5.pdf

Large deviation properties of weakly interacting processes via weak convergence methods

Two-hybrid protein interaction screening in yeast was used to identify proteins that interact with the HBx nonstructural protein of hepatitis B virus (HBV). A new human member of the proteasome alpha-subunit family was obtained. Its protein sequence closely resembles the 28 kD subunits from other organisms. The interaction with HBx was abolished by a two amino-acid insertion behind position 128 in HBx, in a region previously found to be essential for its transcriptional transactivation function. These data support a model of HBx acting indirectly on transcriptional processes. By binding to a specific proteasome alpha-subunit, HBx might interfere with degradative processes, thereby enhancing the half-life of different transcription factors and other nuclear regulatory proteins. Interaction with the Hu 28k proteasome subunit could thus provide a unifying explanation for the markedly pleiotropic effects of HBx.

HBx protein of hepatitis B virus interacts with the C-terminal portion of a novel human proteasome alpha-subunit

The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Ito processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Ito processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm.

/pdf/on-the-moments-of-the-modulus-of-continuity-of-ito-processes-41h6xxu80y.pdf

On the Moments of the Modulus of Continuity of Itô Processes

We study large deviation properties of systems of weakly interacting particles modeled by It\^{o} stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean-Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay.

Markus Fischer

Papers

On the connection between symmetric $N$-player games and mean field games

Large deviation properties of weakly interacting processes via weak convergence methods

HBx protein of hepatitis B virus interacts with the C-terminal portion of a novel human proteasome alpha-subunit

On the Moments of the Modulus of Continuity of Itô Processes

Large deviation properties of weakly interacting processes via weak convergence methods