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

The organization of behavior: A neuropsychological theory

Information Theory and Reliable Communication

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An Introduction To The Theory Of Point Processes

The Foundations of Statistics By Prof. Leonard J. Savage. (Wiley Publications in Statistics.) Pp. xv + 294. (New York; John Wiley and Sons, Inc.; London: Chapman and Hall, Ltd., 1954.) 48s. net.

Probability and Statistical Inference

The aim of this paper is to provide a new method for the detection of either favored or avoided distances between genomic events along DNA sequences. These events are modeled by a Hawkes’ process. The biological problem is actually complex enough to need a non asymptotic penalized model selection approach. We provide a theoretical penalty that satisfies an oracle inequality even for quite complex families of models. The consecutive theoretical estimator is

/pdf/adaptive-estimation-for-hawkes-processes-application-to-4gcr14c922.pdf

Adaptive estimation for Hawkes processes; application to genome analysis

The aim of this paper is to provide a new method for the detection of either favored or avoided distances between genomic events along DNA sequences. These events are modeled by a Hawkes process. The biological problem is actually complex enough to need a nonasymptotic penalized model selection approach. We provide a theoretical penalty that satisfies an oracle inequality even for quite complex families of models. The consecutive theoretical estimator is shown to be adaptive minimax for H\"{o}lderian functions with regularity in $(1/2,1]$: those aspects have not yet been studied for the Hawkes' process. Moreover, we introduce an efficient strategy, named Islands, which is not classically used in model selection, but that happens to be particularly relevant to the biological question we want to answer. Since a multiplicative constant in the theoretical penalty is not computable in practice, we provide extensive simulations to find a data-driven calibration of this constant. The results obtained on real genomic data are coherent with biological knowledge and eventually refine them.

/pdf/adaptive-estimation-for-hawkes-processes-application-to-3lgia10ihg.pdf

Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive $\ell_1$-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Non-asymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activities inference, we finally lead a simulation study for multivariate Hawkes processes and compare our methodology with the {\it adaptive Lasso procedure} proposed by Zou in \cite{Zou}. We observe an excellent behavior of our procedure with respect to the problem of supports recovery. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of \cite{Zou}, our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.

/pdf/lasso-and-probabilistic-inequalities-for-multivariate-point-r3porjq61n.pdf

Lasso and probabilistic inequalities for multivariate point processes

In this paper, we establish oracle inequalities for penalized projection estimators of the intensity of an inhomogeneous Poisson process. We study consequently the adaptive properties of penalized projection estimators. At first we provide lower bounds for the minimax risk over various sets of smoothness for the intensity and then we prove that our estimators achieve these lower bounds up to some constants. The crucial tools to obtain the oracle inequalities are new concentration inequalities for suprema of integral functionals of Poisson processes which are analogous to Talagrand's inequalities for empirical processes.

/pdf/adaptive-estimation-of-the-intensity-of-inhomogeneous-3ge7eo3e8e.pdf

Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities

Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive $\ell_1$-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Nonasymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activity inference, we finally carry out a simulation study for multivariate Hawkes processes and compare our methodology with the adaptive Lasso procedure proposed by Zou in (J. Amer. Statist. Assoc. 101 (2006) 1418-1429). We observe an excellent behavior of our procedure. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of (J. Amer. Statist. Assoc. 101 (2006) 1418-1429), our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.

Patricia Reynaud-Bouret

Papers

Adaptive estimation for Hawkes processes; application to genome analysis

Adaptive estimation for Hawkes processes; application to genome analysis

Lasso and probabilistic inequalities for multivariate point processes

Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities

Lasso and probabilistic inequalities for multivariate point processes