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

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Stochastic equations in infinite dimensions

The flow of homogeneous fluids through porous media: analogies with other physical problems

The Keller-Segel partial differential equation is a two-dimensional model for chemotaxis. When the total mass of the initial density is one, it is known to exhibit blow-up in finite time as soon as the sensitivity $\chi$ of bacteria to the chemo-attractant is larger than $8\pi$. We investigate its approximation by a system of $N$ two-dimensional Brownian particles interacting through a singular attractive kernel in the drift term. In the very subcritical case $\chi\textless{}2\pi$, the diffusion strongly dominates this singular drift: we obtain existence for the particle system and prove that its flow of empirical measures converges, as $N\to\infty$ and up to extraction of a subsequence, to a weak solution of the Keller-Segel equation. We also show that for any $N\ge 2$ and any value of $\chi\textgreater{}0$, pairs of particles do collide with positive probability: the singularity of the drift is indeed visited. Nevertheless, when $\chi\textless{}2\pi N$, it is possible to control the drift and obtain existence of the particle system until the first time when at least three particles collide. We check that this time is a.s. infinite, so that global existence holds for the particle system, if and only if $\chi\leq 8\pi(N-2)/(N-1)$. Finally, we remark that in the system with $N=2$ particles, the difference between the two positions provides a natural two-dimensional generalization of Bessel processes, which we study in details.

Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes

Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids

Motivated by the approximation of Martingale Optimal Transport problems, we
study sampling methods preserving the convex order for two probability measures
$\mu$ and $\nu$ on $\mathbb{R}^d$, with $\nu$ dominating $\mu$. When
$(X_i)_{1\le i\le I}$ (resp. $(Y_j)_{1\le j\le J}$) are i.i.d. according $\mu$
(resp. $\nu$), the empirical measures $\mu_I$ and $\nu_J$ are not in the convex
order. We investigate modifications of $\mu_I$ (resp. $\nu_J$) smaller than
$\nu_J$ (resp. greater than $\mu_I$) in the convex order and weakly converging
to $\mu$ (resp. $\nu$) as $I,J\to\infty$. In dimension 1, according to Kertz
and R\"osler (1992), the set of probability measures with a finite first order
moment is a lattice for the increasing and the decreasing convex orders. From
this result, we can define $\mu\vee\nu$ (resp. $\mu\wedge\nu$) that is greater
than $\mu$ (resp. smaller than $\nu$) in the convex order. We give efficient
algorithms permitting to compute $\mu\vee\nu$ and $\mu\wedge\nu$ when $\mu$ and
$\nu$ are convex combinations of Dirac masses. In general dimension, when $\mu$
and $\nu$ have finite moments of order $\rho\ge 1$, we define the projection
$\mu\curlywedge_\rho \nu$ (resp. $\mu\curlyvee_\rho\nu$) of $\mu$ (resp. $\nu$)
on the set of probability measures dominated by $\nu$ (resp. larger than $\mu$)
in the convex order for the Wasserstein distance with index $\rho$. When
$\rho=2$, $\mu_I\curlywedge_2 \nu_J$ can be computed efficiently by solving a
quadratic optimization problem with linear constraints. It turns out that, in
dimension 1, the projections do not depend on $\rho$ and their quantile
functions are explicit, which leads to efficient algorithms for convex
combinations of Dirac masses. Last, we illustrate by numerical experiments the
resulting sampling methods that preserve the convex order and their application
to approximate Martingale Optimal Transport problems.

Sampling of probability measures in the convex order by Wasserstein projection

Adaptive Monte Carlo methods are very efficient techniques designed to tune simulation estimators on-line. In this work, we present an alternative to stochastic approximation to tune the optimal change of measure in the context of importance sampling for normal random vectors. Unlike stochastic approximation, which requires very fine tuning in practice, we propose to use sample average approximation and deterministic optimization techniques to devise a robust and fully automatic variance reduction methodology. The same samples are used in the sample optimization of the importance sampling parameter and in the Monte Carlo computation of the expectation of interest with the optimal measure computed in the previous step. We prove that this highly dependent Monte Carlo estimator is convergent and satisfies a central limit theorem with the optimal limiting variance. Numerical experiments confirm the performance of this estimator: in comparison with the crude Monte Carlo method, the computation time needed to achieve a given precision is divided by a factor between 3 and 15.

/pdf/robust-adaptive-importance-sampling-for-normal-random-4vwm9abroe.pdf

Robust adaptive importance sampling for normal random vectors.

The micro–macro simulations of polymeric fluids couple the mass and momentum conservation equations at the macroscopic level, with a stochastic differential equation which models the evolution of the polymer configurations at the microscopic level (Brownian dynamics simulation). Accordingly, the system is discretized by a finite element method coupled with a Monte Carlo method. All the discrete variables are random, and the accuracy of the result highly depends on the variance of these random variables. We give here some elements of numerical analysis on the crucial issue of variance reduction in order to get results of a better quality for a given computational cost. The present analytical study only deals with a one-dimensional case, but nevertheless gives a track for computational strategies that may apply to the more physically relevant two- and three-dimensional cases.

Benjamin Jourdain

Papers

Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes

Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids

Sampling of probability measures in the convex order by Wasserstein projection

Robust adaptive importance sampling for normal random vectors.

On a variance reduction technique for micro–macro simulations of polymeric fluids