Mathieu Gerber, Nicolas Chopin, Nick Whiteley.

Source: The Annals of Statistics, Volume 47, Number 4, 2236--2260.

Abstract:
We study convergence and convergence rates for resampling schemes. Our first main result is a general consistency theorem based on the notion of negative association, which is applied to establish the almost sure weak convergence of measures output from Kitagawa’s [ J. Comput. Graph. Statist. 5 (1996) 1–25] stratified resampling method. Carpenter, Ckiffird and Fearnhead’s [ IEE Proc. Radar Sonar Navig. 146 (1999) 2–7] systematic resampling method is similar in structure but can fail to converge depending on the order of the input samples. We introduce a new resampling algorithm based on a stochastic rounding technique of [In 42nd IEEE Symposium on Foundations of Computer Science ( Las Vegas , NV , 2001) (2001) 588–597 IEEE Computer Soc.], which shares some attractive properties of systematic resampling, but which exhibits negative association and, therefore, converges irrespective of the order of the input samples. We confirm a conjecture made by [ J. Comput. Graph. Statist. 5 (1996) 1–25] that ordering input samples by their states in $mathbb{R}$ yields a faster rate of convergence; we establish that when particles are ordered using the Hilbert curve in $mathbb{R}^{d}$, the variance of the resampling error is ${scriptstylemathcal{O}}(N^{-(1+1/d)})$ under mild conditions, where $N$ is the number of particles. We use these results to establish asymptotic properties of particle algorithms based on resampling schemes that differ from multinomial resampling.

Takashi Owada, Omer Bobrowski.

Source: Bernoulli, Volume 26, Number 3, 2275--2310.

Abstract:
In this paper, we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of topological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to topological cycles generated by random points far away from the bulk of other points, when the support is unbounded. We establish weak convergence results for persistence diagrams – a point process representation for persistent homology, where each topological cycle is represented by its $({mathit{birth},mathit{death}})$ coordinates. In this work, we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. Using this framework, we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a shorter lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.

Emanuele Dolera, Stefano Favaro.

Source: Bernoulli, Volume 26, Number 2, 1294--1322.

Abstract:
Given a sequence ${X_{n}}_{ngeq 1}$ of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that $frac{1}{n}sum_{i=1}^{n}X_{i}stackrel{a.s.}{longrightarrow }Y$ for a suitable random variable $Y:Omega ightarrow [0,1]$ satisfying $mathsf{P}[X_{1}=x_{1},dots ,X_{n}=x_{n}|Y]=Y^{sum_{i=1}^{n}x_{i}}(1-Y)^{n-sum_{i=1}^{n}x_{i}}$. In this paper, we study the rate of convergence in law of $frac{1}{n}sum_{i=1}^{n}X_{i}$ to $Y$ under the Kolmogorov distance. After showing that a rate of the type of $1/n^{alpha }$ can be obtained for any index $alpha in (0,1]$, we find a sufficient condition on the distribution of $Y$ for the achievement of the optimal rate of convergence, that is $1/n$. Besides extending and strengthening recent results under the weaker Wasserstein distance, our main result weakens the regularity hypotheses on $Y$ in the context of the Hausdorff moment problem.

Yating Liu, Gilles Pagès.

Source: Bernoulli, Volume 26, Number 2, 1171--1204.

Abstract:
We establish conditions to characterize probability measures by their $L^{p}$-quantization error functions in both $mathbb{R}^{d}$ and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic (convergence for the $L^{p}$-Wasserstein distance). We first propose a criterion on the quantization level $N$, valid for any norm on $mathbb{R}^{d}$ and any order $p$ based on a geometrical approach involving the Voronoï diagram. Then, we prove that in the $L^{2}$-case on a (separable) Hilbert space, the condition on the level $N$ can be reduced to $N=2$, which is optimal. More quantization based characterization cases in dimension 1 and a discussion of the completeness of a distance defined by the quantization error function can be found at the end of this paper.

Jie Yen Fan, Kais Hamza, Peter Jagers, Fima Klebaner.

Source: Bernoulli, Volume 26, Number 2, 893--926.

Abstract:
We consider a family of general branching processes with reproduction parameters depending on the age of the individual as well as the population age structure and a parameter $K$, which may represent the carrying capacity. These processes are Markovian in the age structure. In a previous paper ( Proc. Steklov Inst. Math. 282 (2013) 90–105), the Law of Large Numbers as $K o infty $ was derived. Here we prove the central limit theorem, namely the weak convergence of the fluctuation processes in an appropriate Skorokhod space. We also show that the limit is driven by a stochastic partial differential equation.

Jing Lei.

Source: Bernoulli, Volume 26, Number 1, 767--798.

Abstract:
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions.

Tobias Zwingmann, Hajo Holzmann.

Source: Bernoulli, Volume 26, Number 1, 323--351.

Abstract:
We show weak convergence of quantile and expectile processes to Gaussian limit processes in the space of bounded functions endowed with an appropriate semimetric which is based on the concepts of epi- and hypo- convergence as introduced in A. Bücher, J. Segers and S. Volgushev (2014), ‘ When Uniform Weak Convergence Fails: Empirical Processes for Dependence Functions and Residuals via Epi- and Hypographs ’, Annals of Statistics 42 . We impose assumptions for which it is known that weak convergence with respect to the supremum norm generally fails to hold. For quantiles, we consider stationary observations, where the marginal distribution function is assumed to be strictly increasing and continuous except for finitely many points and to admit strictly positive – possibly infinite – left- and right-sided derivatives. For expectiles, we focus on independent and identically distributed (i.i.d.) observations. Only a finite second moment and continuity at the boundary points but no further smoothness properties of the distribution function are required. We also show consistency of the bootstrap for this mode of convergence in the i.i.d. case for quantiles and expectiles.

The International Competition Network advanced convergence through important initiatives on international enforcement cooperation and investigative processes in competition cases.

The International Competition Network (ICN) adopted new recommended practices for predatory pricing analysis and competition assessment, and approved new work product on international merger enforcement cooperation, confidentiality protections during investigations, leniency policy and digital evidence gathering.

Across the OECD, GDP per capita is converging. In contrast, regional disparities – or differences in GDP per capita across jurisdictions – are rising, mainly as a result of widening productivity differences. Fiscal decentralisation could help reduce them again. According to new OECD research, assigning more ownsource revenue to sub-national governments dampens regional GDP disparities and underpins regional convergence.

While Korea remains one of the fastest-growing OECD economies, its potential growth rate per capita is projected to decelerate from around 4% during the current decade to around 2¼ per cent during the 2030s.

This paper analyses convergence in per capita gross regional product of Russia’s regions during the period 1995-2010, when regional data are available.

The Czech economy is finally coming out of a prolonged recession but must take further steps to speed up income convergence towards the euro area countries, according to the OECD’s latest Economic Survey of the Czech Republic.

This paper explores the role of managerial capital and business research and development (R&D) in fostering multifactor productivity (MFP) convergence in a panel of 42 countries.

The underlying strengths of the Estonian economy have helped it bounce back from the crisis, but some challenges remain to finding a steeper, more inclusive and more sustainable growth path, according to the latest OECD Economic Survey of Estonia.

The Czech Republic needs new reforms to boost productivity, improve economic growth and accelerate convergence toward the levels of income and well-being seen in the most advanced European countries, according to a new OECD report.

GDP per capita in Lithuania rose from one third to two thirds of the OECD average level between 1995 and 2014, despite internal and external crises. Productivity catch-up was critical to this process, although the level of labour productivity also remains around one-third below the OECD average.

Although Lithuania’s growth has been impressive, inequality is high, the risk of poverty is one of the highest of European countries, and life expectancy is comparatively low and strongly dependent on socio-economic background.

GDP per capita in Lithuania rose from one third to two thirds of the OECD average level between 1995 and 2014, despite internal and external crises. Productivity catch-up was critical to this process, although the level of labour productivity also remains around one-third below the OECD average.

Although Lithuania’s growth has been impressive, inequality is high, the risk of poverty is one of the highest of European countries, and life expectancy is comparatively low and strongly dependent on socio-economic background.

Boosting national productivity to sustain the convergence process towards OECD countries living standards will hinge on creating the right conditions for domestic firms to thrive and become more innovative and productive, while maintaining the long-standing commitment to open international markets and investment.

Across the OECD, GDP per capita is converging. In contrast, regional disparities – or differences in GDP per capita across jurisdictions – are rising, mainly as a result of widening productivity differences. Fiscal decentralisation could help reduce them again. According to new OECD research, assigning more ownsource revenue to sub-national governments dampens regional GDP disparities and underpins regional convergence.

Over the past two decades, the income level of the Czech Republic has converged considerably towards the OECD average.

Over the past two decades, the income level of the Czech Republic has converged considerably towards the OECD average.

The paper describes revisions to the trend labour efficiency component of the production function underpinning long-term economic scenarios.

Successful implementation of economic reforms has boosted the Latvian economy, leading to strong growth, rising wages and solid public finances. Further policy action is now needed to accelerate productivity growth, create jobs, drive down poverty, improve living standards and ensure that everyone benefits from more inclusive growth, according to a new report from the OECD.

Tunisia is firmly committed to a process of democratisation that needs underpinning by economic reforms in order to guarantee an improved standard of living for all Tunisians.

Across the OECD, GDP per capita is converging. In contrast, regional disparities – or differences in GDP per capita across jurisdictions – are rising, mainly as a result of widening productivity differences. Fiscal decentralisation could help reduce them again. According to new OECD research, assigning more ownsource revenue to sub-national governments dampens regional GDP disparities and underpins regional convergence.

The growth pattern of OECD countries and their sub-national entities is puzzling. Between-country differences in GDP per capita are declining, yet the differences across jurisdictions within those countries tend to rise.

Hayden Library - PN6725.M34 2017

Online Resource

Meikle, Graham, 1965-

Hayden Library - K4240.H86 2018

Rotch Library - HT169.S55 B85 2019