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

Convergence of Probability Measures

Weak Convergence and Empirical Processes - A. W. van der Vaart; J. A. Wellner.

1 Introduction.- 2 Experiments, Deficiencies, Distances v.- 2.1 Comparing Risk Functions.- 2.2 Deficiency and Distance between Experiments.- 2.3 Likelihood Ratios and Blackwell's Representation.- 2.4 Further Remarks on the Convergence of Distri butions of Likelihood Ratios.- 2.5 Historical Remarks.- 3 Contiguity - Hellinger Transforms.- 3.1 Contiguity.- 3.2 Hellinger Distances, Hellinger Transforms.- 3.3 Historical Remarks.- 4 Gaussian Shift and Poisson Experiments.- 4.1 Introduction.- 4.2 Gaussian Experiments.- 4.3 Poisson Experiments.- 4.4 Historical Remarks.- 5 Limit Laws for Likelihood Ratios.- 5.1 Introduction.- 5.2 Auxiliary Results.- 5.2.1 Lindeberg's Procedure.- 5.2.2 Levy Splittings.- 5.2.3 Paul Levy's Symmetrization Inequalities.- 5.2.4 Conditions for Shift-Compactness.- 5.2.5 A Central Limit Theorem for Infinitesimal Arrays.- 5.2.6 The Special Case of Gaussian Limits.- 5.2.7 Peano Differentiable Functions.- 5.3 Limits for Binary Experiments.- 5.4 Gaussian Limits.- 5.5 Historical Remarks.- 6 Local Asymptotic Normality.- 6.1 Introduction.- 6.2 Locally Asymptotically Quadratic Families.- 6.3 A Method of Construction of Estimates.- 6.4 Some Local Bayes Properties.- 6.5 Invariance and Regularity.- 6.6 The LAMN and LAN Conditions.- 6.7 Additional Remarks on the LAN Conditions.- 6.8 Wald's Tests and Confidence Ellipsoids.- 6.9 Possible Extensions.- 6.10 Historical Remarks.- 7 Independent, Identically Distributed Observations.- 7.1 Introduction.- 7.2 The Standard i.i.d. Case: Differentiability in Quadratic Mean.- 7.3 Some Examples.- 7.4 Some Nonparametric Considerations.- 7.5 Bounds on the Risk of Estimates.- 7.6 Some Cases Where the Number of Observations Is Random.- 7.7 Historical Remarks.- 8 On Bayes Procedures.- 8.1 Introduction.- 8.2 Bayes Procedures Behave Nicely.- 8.3 The Bernstein-von Mises Phenomenon.- 8.4 A Bernstein-von Mises Result for the i.i.d. Case.- 8.5 Bayes Procedures Behave Miserably.- 8.6 Historical Remarks.- Author Index.

Asymptotics in Statistics–Some Basic Concepts

1. Preliminaries 2. Two-dimensional Navier-Stokes equations 3. Uniqueness of stationary measure and mixing 4. Ergodicity and limiting theorems 5. Inviscid limit 6. Miscellanies 7. Appendix 8. Solutions to some exercises.

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Mathematics of Two-Dimensional Turbulence

The pseudomarginal algorithm is a Metropolis–Hastings‐type scheme which samples asymptotically from a target probability density when we can only estimate unbiasedly an unnormalized version of it. In a Bayesian context, it is a state of the art posterior simulation technique when the likelihood function is intractable but can be estimated unbiasedly by using Monte Carlo samples. However, for the performance of this scheme not to degrade as the number T of data points increases, it is typically necessary for the number N of Monte Carlo samples to be proportional to T to control the relative variance of the likelihood ratio estimator appearing in the acceptance probability of this algorithm. The correlated pseudomarginal method is a modification of the pseudomarginal method using a likelihood ratio estimator computed by using two correlated likelihood estimators. For random‐effects models, we show under regularity conditions that the parameters of this scheme can be selected such that the relative variance of this likelihood ratio estimator is controlled when N increases sublinearly with T and we provide guidelines on how to optimize the algorithm on the basis of a non‐standard weak convergence analysis. The efficiency of computations for Bayesian inference relative to the pseudomarginal method empirically increases with T and exceeds two orders of magnitude in some examples.

The correlated pseudomarginal method

This paper deals with empirical processes of the type $$C_{n}(B)=\sqrt{n}\{\mu_{n}(B)-P(X_{n+1}\in B\mid X_{1},\ldots,X_{n})\},$$ where $(X_n)$ is a sequence of random variables and $μ_n=(1/n)∑_{i=1}^nδ_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029–2052]). By such conditions, in some relevant situations, one obtains that $\sup_{B}|C_{n}(B)|\stackrel{P}{\rightarrow}0$ or even that $\sqrt{n}\sup_{B}|C_{n}(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.

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Rate of convergence of predictive distributions for dependent data

. The Hirsch index (commonly referred to as h-index) is a bibliometric indicator which is widely recognized as effective for measuring the scientific production of a scholar since it summarizes size and impact of the research output. In a formal setting, the h-index is actually an empirical functional of the distribution of the citation counts received by the scholar. Under this approach, the asymptotic theory for the empirical h-index has been recently exploited when the citation counts follow a continuous distribution and, in particular, variance estimation has been considered for the Pareto-type and the Weibull-type distribution families. However, in bibliometric applications, citation counts display a distribution supported by the integers. Thus, we provide general properties for the empirical h-index under the small- and large-sample settings. In addition, we also introduce consistent non-parametric variance estimation, which allows for the implementation of large-sample set estimation for the theoretical h-index.

Statistical Analysis of the Hirsch Index

In a Bayesian framework, to make predictions on a sequence $X_{1},X_{2},\ldots $ of random observations, the inferrer needs to assign the predictive distributions $\sigma _{n}(\cdot )=P(X_{n+1}\in \cdot \mid X_{1},\ldots,X_{n})$. In this paper, we propose to assign $\sigma _{n}$ directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be assessed. The data sequence $(X_{n})$ is assumed to be conditionally identically distributed (c.i.d.) in the sense of (Ann. Probab. 32 (2004) 2029–2052). To realize this programme, a class $\Sigma $ of predictive distributions is introduced and investigated. Such a $\Sigma $ is rich enough to model various real situations and $(X_{n})$ is actually c.i.d. if $\sigma _{n}$ belongs to $\Sigma $. Furthermore, when a new observation $X_{n+1}$ becomes available, $\sigma _{n+1}$ can be obtained by a simple recursive update of $\sigma _{n}$. If $\mu $ is the a.s. weak limit of $\sigma _{n}$, conditions for $\mu $ to be a.s. discrete are provided as well.

A class of models for Bayesian predictive inference

https://www.econstor.eu/bitstream/10419/95302/1/q106.pdf

Central limit theorems for multicolor urns with dominated colors

Let $(X_n)$ be any sequence of random variables adapted to a filtration $(\mathcal{G}_n)$. Define $a_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid\mathcal{G}_n\bigr)$, $b_n=\frac{1}{n}\sum_{i=0}^{n-1}a_i$, and $\mu_n=\frac{1}{n}\,\sum_{i=1}^n\delta_{X_i}$. Convergence in distribution of the empirical processes $$ B_n=\sqrt{n}\,(\mu_n-b_n)\quad\text{and}\quad C_n=\sqrt{n}\,(\mu_n-a_n)$$ is investigated under uniform distance. If $(X_n)$ is conditionally identically distributed, convergence of $B_n$ and $C_n$ is studied according to Meyer-Zheng as well. Some CLTs, both uniform and non uniform, are proved. In addition, various examples and a characterization of conditionally identically distributed sequences are given.

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Luca Pratelli

Papers

Rate of convergence of predictive distributions for dependent data

Statistical Analysis of the Hirsch Index

A class of models for Bayesian predictive inference

Central limit theorems for multicolor urns with dominated colors

Limit theorems for empirical processes based on dependent data