## Recursos de colección

Bernoulli

1. #### On the Poisson equation for Metropolis–Hastings chains

Mijatović, Aleksandar; Vogrinc, Jure
This paper defines an approximation scheme for a solution of the Poisson equation of a geometrically ergodic Metropolis–Hastings chain $\Phi$. The scheme is based on the idea of weak approximation and gives rise to a natural sequence of control variates for the ergodic average $S_{k}(F)=(1/k)\sum_{i=1}^{k}F(\Phi_{i})$, where $F$ is the force function in the Poisson equation. The main results show that the sequence of the asymptotic variances (in the CLTs for the control-variate estimators) converges to zero and give a rate of this convergence. Numerical examples in the case of a double-well potential are discussed.

2. #### On the local semicircular law for Wigner ensembles

Götze, Friedrich; Naumov, Alexey; Tikhomirov, Alexander; Timushev, Dmitry
We consider a random symmetric matrix $\mathbf{X}=[X_{jk}]_{j,k=1}^{n}$ with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that $\mathbb{E}|X_{11}|^{4+\delta}=:\mu_{4+\delta}<\infty$ for some $\delta>0$. The aim of this paper is to significantly extend a recent result of the authors Götze, Naumov and Tikhomirov (2015) and show that with high probability the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-\frac{1}{2}}\mathbf{X}$ and Wigner’s semicircle law is of order $(nv)^{-1}\log n$, where $v$ denotes the distance to the real line in the complex plane. We apply this result to the...

3. #### M-estimators of location for functional data

Sinova, Beatriz; González-Rodríguez, Gil; Van Aelst, Stefan
M-estimators of location are widely used robust estimators of the center of univariate or multivariate real-valued data. This paper aims to study M-estimates of location in the framework of functional data analysis. To this end, recent developments for robust nonparametric density estimation by means of M-estimators are considered. These results can also be applied in the context of functional data analysis and allow to state conditions for the existence and uniqueness of location M-estimates in this setting. Properties of these functional M-estimators are investigated. In particular, their consistency is shown and robustness is studied by means of their breakdown point...

4. #### Schwarz type model comparison for LAQ models

Eguchi, Shoichi; Masuda, Hiroki
For model-comparison purpose, we study asymptotic behavior of the marginal quasi-log likelihood associated with a family of locally asymptotically quadratic (LAQ) statistical experiments. Our result entails a far-reaching extension of applicable scope of the classical approximate Bayesian model comparison due to Schwarz, with frequentist-view theoretical foundation. In particular, the proposed statistics can deal with both ergodic and non-ergodic stochastic process models, where the corresponding $M$-estimator may of multi-scaling type and the asymptotic quasi-information matrix may be random. We also deduce the consistency of the multistage optimal-model selection where we select an optimal sub-model structure step by step, so that computational...

5. #### Equilibrium of the interface of the grass-bushes-trees process

Andjel, Enrique; Mountford, Thomas; Valesin, Daniel
We consider the grass-bushes-trees process, which is a two-type contact process in which one of the types is dominant. Individuals of the dominant type can give birth on empty sites and sites occupied by non-dominant individuals, whereas non-dominant individuals can only give birth at empty sites. We study the shifted version of this process so that it is ‘seen from the rightmost dominant individual’ (which is well defined if the process occurs in an appropriate subset of the configuration space); we call this shifted process the grass-bushes-trees interface (GBTI) process. The set of stationary distributions of the GBTI process is...

6. #### Dynamics of an adaptive randomly reinforced urn

Aletti, Giacomo; Ghiglietti, Andrea; Vidyashankar, Anand N.
Adaptive randomly reinforced urn (ARRU) is a two-color urn model where the updating process is defined by a sequence of non-negative random vectors $\{(D_{1,n},D_{2,n});n\geq1\}$ and randomly evolving thresholds which utilize accruing statistical information for the updates. Let $m_{1}=E[D_{1,n}]$ and $m_{2}=E[D_{2,n}]$. In this paper, we undertake a detailed study of the dynamics of the ARRU model. First, for the case $m_{1}\neq m_{2}$, we establish $L_{1}$ bounds on the increments of the urn proportion, that is, the proportion of ball colors in the urn, at fixed and increasing times under very weak assumptions on the random threshold sequences. As a consequence, we...

7. #### On optimality of empirical risk minimization in linear aggregation

In the first part of this paper, we show that the small-ball condition, recently introduced by (J. ACM 62 (2015) Art. 21, 25), may behave poorly for important classes of localized functions such as wavelets, piecewise polynomials or for trigonometric polynomials, in particular leading to suboptimal estimates of the rate of convergence of ERM for the linear aggregation problem. In a second part, we recover optimal rates of convergence for the excess risk of ERM when the dictionary is made of trigonometric functions. Considering the bounded case, we derive the concentration of the excess risk around a single point, which...

8. #### The function-indexed sequential empirical process under long-range dependence

Buchsteiner, Jannis
Let $(\boldsymbol{X}_{j})_{j\geq1}$ be a multivariate long-range dependent Gaussian process. We study the asymptotic behavior of the corresponding sequential empirical process indexed by a class of functions. If some entropy condition is satisfied we have weak convergence to a linear combination of Hermite processes.

9. #### Bayesian non-parametric inference for $\Lambda$-coalescents: Posterior consistency and a parametric method

Koskela, Jere; Jenkins, Paul A.; Spanò, Dario
We investigate Bayesian non-parametric inference of the $\Lambda$-measure of $\Lambda$-coalescent processes with recurrent mutation, parametrised by probability measures on the unit interval. We give verifiable criteria on the prior for posterior consistency when observations form a time series, and prove that any non-trivial prior is inconsistent when all observations are contemporaneous. We then show that the likelihood given a data set of size $n\in \mathbb{N}$ is constant across $\Lambda$-measures whose leading $n-2$ moments agree, and focus on inferring truncated sequences of moments. We provide a large class of functionals which can be extremised using finite computation given a credible region...

10. #### A general approach to posterior contraction in nonparametric inverse problems

Knapik, Bartek; Salomond, Jean-Bernard
In this paper, we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive contraction rates for the parameter of interest from contraction rates of the related direct problem of estimating transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive contraction rates for priors that are not related to the singular value decomposition of the operator. We apply our result to several examples of linear inverse problems, both in the white...

11. #### On parameter estimation of hidden telegraph process

Khasminskii, Rafail Z.; Kutoyants, Yury A.
The problem of parameter estimation is considered for the two-state telegraph process, observed in the white Gaussian observation noise. An online one-step Maximum Likelihood Estimator process is constructed, using a preliminary Method of Moments Estimator. The obtained estimation procedure is shown to be asymptotically normal and asymptotically efficient in the large sample regime.

12. #### Deviation of polynomials from their expectations and isoperimetry

Arutyunyan, Lavrentin M.; Kosov, Egor D.
The article is divided into two parts. In the first part, we study the deviation of a polynomial from its mathematical expectation. This deviation can be estimated from above by Carbery–Wright inequality, so we investigate estimates of the deviation from below. We obtain such type estimates in two different cases: for Gaussian measures and a polynomial of an arbitrary degree and for an arbitrary log-concave measure but only for polynomials of the second degree. In the second part, we deal with the isoperimetric inequality and the Poincaré inequality for probability measures on the real line that are images of the...

13. #### Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs

Bossy, Mireille; Olivero, Héctor
In this paper, we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE $X_{t}=x_{0}+\int_{0}^{t}{b(X_{s})\,ds}+\int_{0}^{t}{\sigma\vert X_{s}\vert^{\alpha}\,dW_{s}},\qquad x_{0}>0,\sigma>0,\alpha\in[\frac{1}{2},1).$ Assuming $b(0)/\sigma^{2}$ big enough, and $b$ smooth, we prove a strong rate of convergence of order one, recovering the classical result of Milstein for SDEs with smooth diffusion coefficient. In contrast with other recent results, our proof does not relies on Lamperti transformation, and it can be applied to a wide class of drift functions. On the downside, our hypothesis on the critical parameter value $b(0)/\sigma^{2}$ is more restrictive than others...

14. #### When do wireless network signals appear Poisson?

Keeler, H. Paul; Ross, Nathan; Xia, Aihua
We consider the point process of signal strengths from transmitters in a wireless network observed from a fixed position under models with general signal path loss and random propagation effects. We show via coupling arguments that under general conditions this point process of signal strengths can be well-approximated by an inhomogeneous Poisson or a Cox point processes on the positive real line. We also provide some bounds on the total variation distance between the laws of these point processes and both Poisson and Cox point processes. Under appropriate conditions, these results support the use of a spatial Poisson point process...

15. #### Tree formulas, mean first passage times and Kemeny’s constant of a Markov chain

Pitman, Jim; Tang, Wenpin
This paper offers some probabilistic and combinatorial insights into tree formulas for the Green function and hitting probabilities of Markov chains on a finite state space. These tree formulas are closely related to loop-erased random walks by Wilson’s algorithm for random spanning trees, and to mixing times by the Markov chain tree theorem. Let $m_{ij}$ be the mean first passage time from $i$ to $j$ for an irreducible chain with finite state space $S$ and transition matrix $(p_{ij};i,j\in S)$. It is well known that $m_{jj}=1/\pi_{j}=\Sigma^{(1)}/\Sigma_{j}$, where $\pi$ is the stationary distribution for the chain, $\Sigma_{j}$ is the tree sum, over...

16. #### Finite sample properties of the mean occupancy counts and probabilities

Decrouez, Geoffrey; Grabchak, Michael; Paris, Quentin
For a probability distribution $P$ on an at most countable alphabet $\mathcal{A}$, this article gives finite sample bounds for the expected occupancy counts $\mathbb{E}K_{n,r}$ and probabilities $\mathbb{E}M_{n,r}$. Both upper and lower bounds are given in terms of the counting function $\nu$ of $P$. Special attention is given to the case where $\nu$ is bounded by a regularly varying function. In this case, it is shown that our general results lead to an optimal-rate control of the expected occupancy counts and probabilities with explicit constants. Our results are also put in perspective with Turing’s formula and recent concentration bounds to deduce...

17. #### Strong consistency of multivariate spectral variance estimators in Markov chain Monte Carlo

Vats, Dootika; Flegal, James M.; Jones, Galin L.
Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. The Monte Carlo error in estimation has an asymptotic normal distribution whose multivariate nature has so far been ignored in the MCMC community. We present a class of multivariate spectral variance estimators for the asymptotic covariance matrix in the Markov chain central limit theorem and provide conditions for strong consistency. We examine the finite sample properties of the multivariate spectral variance estimators and its eigenvalues in the context of a vector autoregressive process of order 1.

18. #### Extrema of rescaled locally stationary Gaussian fields on manifolds

Qiao, Wanli; Polonik, Wolfgang
Given a class of centered Gaussian random fields $\{X_{h}(s),s\in\mathbb{R}^{n},h\in(0,1]\}$, define the rescaled fields $\{Z_{h}(t)=X_{h}(h^{-1}t),t\in\mathcal{M}\}$, where $\mathcal{M}$ is a compact Riemannian manifold. Under the assumption that the fields $Z_{h}(t)$ satisfy a local stationary condition, we study the limit behavior of the extreme values of these rescaled Gaussian random fields, as $h$ tends to zero. Our main result can be considered as a generalization of a classical result of Bickel and Rosenblatt (Ann. Statist. 1 (1973) 1071–1095), and also of results by Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367–379).

19. #### On Gaussian comparison inequality and its application to spectral analysis of large random matrices

Han, Fang; Xu, Sheng; Zhou, Wen-Xin
Recently, Chernozhukov, Chetverikov, and Kato (Ann. Statist. 42 (2014) 1564–1597) developed a new Gaussian comparison inequality for approximating the suprema of empirical processes. This paper exploits this technique to devise sharp inference on spectra of large random matrices. In particular, we show that two long-standing problems in random matrix theory can be solved: (i) simple bootstrap inference on sample eigenvalues when true eigenvalues are tied; (ii) conducting two-sample Roy’s covariance test in high dimensions. To establish the asymptotic results, a generalized $\varepsilon$-net argument regarding the matrix rescaled spectral norm and several new empirical process bounds are developed and of independent...

20. #### Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models

Agapiou, Sergios; Roberts, Gareth O.; Vollmer, Sebastian J.
We provide a general methodology for unbiased estimation for intractable stochastic models. We consider situations where the target distribution can be written as an appropriate limit of distributions, and where conventional approaches require truncation of such a representation leading to a systematic bias. For example, the target distribution might be representable as the $L^{2}$-limit of a basis expansion in a suitable Hilbert space; or alternatively the distribution of interest might be representable as the weak limit of a sequence of random variables, as in MCMC. Our main motivation comes from infinite-dimensional models which can be parameterised in terms of a...

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