## Recursos de colección

Bernoulli

1. #### Uniform dimension results for a family of Markov processes

Sun, Xiaobin; Xiao, Yimin; Xu, Lihu; Zhai, Jianliang
In this paper, we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles in Xiao (In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 (2004) 261–338 Amer. Math. Soc.). As applications, uniform Hausdorff and packing dimension results for certain classes of Lévy processes, stable jump diffusions and non-symmetric stable-type processes are obtained.

2. #### Robust dimension-free Gram operator estimates

Giulini, Ilaria
In this paper, we investigate the question of estimating the Gram operator by a robust estimator from an i.i.d. sample in a separable Hilbert space and we present uniform bounds that hold under weak moment assumptions. The approach consists in first obtaining non-asymptotic dimension-free bounds in finite-dimensional spaces using some PAC-Bayesian inequalities related to Gaussian perturbations of the parameter and then in generalizing the results in a separable Hilbert space. We show both from a theoretical point of view and with the help of some simulations that such a robust estimator improves the behavior of the classical empirical one in...

3. #### Covariance estimation via sparse Kronecker structures

Leng, Chenlei; Pan, Guangming
The problem of estimating covariance matrices is central to statistical analysis and is extensively addressed when data are vectors. This paper studies a novel Kronecker-structured approach for estimating such matrices when data are matrices and arrays. Focusing on matrix-variate data, we present simple approaches to estimate the row and the column correlation matrices, formulated separately via convex optimization. We also discuss simple thresholding estimators motivated by the recent development in the literature. Non-asymptotic results show that the proposed method greatly outperforms methods that ignore the matrix structure of the data. In particular, our framework allows the dimensionality of data to...

4. #### Optimal estimation of a large-dimensional covariance matrix under Stein’s loss

Ledoit, Olivier; Wolf, Michael
This paper introduces a new method for deriving covariance matrix estimators that are decision-theoretically optimal within a class of nonlinear shrinkage estimators. The key is to employ large-dimensional asymptotics: the matrix dimension and the sample size go to infinity together, with their ratio converging to a finite, nonzero limit. As the main focus, we apply this method to Stein’s loss. Compared to the estimator of Stein (Estimation of a covariance matrix (1975); J. Math. Sci. 34 (1986) 1373–1403), ours has five theoretical advantages: (1) it asymptotically minimizes the loss itself, instead of an estimator of the expected loss; (2) it...

5. #### The class of multivariate max-id copulas with $\ell_{1}$-norm symmetric exponent measure

Genest, Christian; Nešlehová, Johanna G.; Rivest, Louis-Paul
Members of the well-known family of bivariate Galambos copulas can be expressed in a closed form in terms of the univariate Fréchet distribution. This formula extends to any dimension and can be used to define a whole new class of tractable multivariate copulas that are generated by suitable univariate distributions. This paper gives necessary and sufficient conditions on the underlying univariate distribution which ensure that the resulting copula exists. It is also shown that these new copulas are in fact dependence structures of certain max-id distributions with $\ell_{1}$-norm symmetric exponent measure. The basic dependence properties of this new class of...

6. #### Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution

Kamatani, Kengo
The purpose of this paper is to introduce a new Markov chain Monte Carlo method and to express its effectiveness by simulation and high-dimensional asymptotic theory. The key fact is that our algorithm has a reversible proposal kernel, which is designed to have a heavy-tailed invariant probability distribution. A high-dimensional asymptotic theory is studied for a class of heavy-tailed target probability distributions. When the number of dimensions of the state space passes to infinity, we will show that our algorithm has a much higher convergence rate than the pre-conditioned Crank–Nicolson (pCN) algorithm and the random-walk Metropolis algorithm.

7. #### Adaptive estimation of high-dimensional signal-to-noise ratios

Verzelen, Nicolas; Gassiat, Elisabeth
We consider the equivalent problems of estimating the residual variance, the proportion of explained variance $\eta$ and the signal strength in a high-dimensional linear regression model with Gaussian random design. Our aim is to understand the impact of not knowing the sparsity of the vector of regression coefficients and not knowing the distribution of the design on minimax estimation rates of $\eta$. Depending on the sparsity $k$ of the vector regression coefficients, optimal estimators of $\eta$ either rely on estimating the vector of regression coefficients or are based on $U$-type statistics. In the important situation where $k$ is unknown, we...

8. #### Large volatility matrix estimation with factor-based diffusion model for high-frequency financial data

Kim, Donggyu; Liu, Yi; Wang, Yazhen
Large volatility matrices are involved in many finance practices, and estimating large volatility matrices based on high-frequency financial data encounters the “curse of dimensionality”. It is a common approach to impose a sparsity assumption on the large volatility matrices to produce consistent volatility matrix estimators. However, due to the existence of common factors, assets are highly correlated with each other, and it is not reasonable to assume the volatility matrices are sparse in financial applications. This paper incorporates factor influence in the asset pricing model and investigates large volatility matrix estimation under the factor price model together with some sparsity...

9. #### Detecting Markov random fields hidden in white noise

Arias-Castro, Ery; Bubeck, Sébastien; Lugosi, Gábor; Verzelen, Nicolas
Motivated by change point problems in time series and the detection of textured objects in images, we consider the problem of detecting a piece of a Gaussian Markov random field hidden in white Gaussian noise. We derive minimax lower bounds and propose near-optimal tests.

10. #### Correlated continuous time random walks and fractional Pearson diffusions

Leonenko, N.N.; Papić, I.; Sikorskii, A.; Šuvak, N.
Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model and Wright–Fisher model. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments. The waiting times are selected from the domain of attraction of a stable law.

11. #### Statistical estimation of the Oscillating Brownian Motion

Lejay, Antoine; Pigato, Paolo
We study the asymptotic behavior of estimators of a two-valued, discontinuous diffusion coefficient in a Stochastic Differential Equation, called an Oscillating Brownian Motion. Using the relation of the latter process with the Skew Brownian Motion, we propose two natural consistent estimators, which are variants of the integrated volatility estimator and take the occupation times into account. We show the stable convergence of the renormalized errors’ estimations toward some Gaussian mixture, possibly corrected by a term that depends on the local time. These limits stem from the lack of ergodicity as well as the behavior of the local time at zero...

12. #### Testing for simultaneous jumps in case of asynchronous observations

Martin, Ole; Vetter, Mathias
This paper proposes a novel test for simultaneous jumps in a bivariate Itô semimartingale when observation times are asynchronous and irregular. Inference is built on a realized correlation coefficient for the squared jumps of the two processes which is estimated using bivariate power variations of Hayashi–Yoshida type without an additional synchronization step. An associated central limit theorem is shown whose asymptotic distribution is assessed using a bootstrap procedure. Simulations show that the test works remarkably well in comparison with the much simpler case of regular observations.

13. #### Small deviations of a Galton–Watson process with immigration

We consider a Galton–Watson process with immigration $(\mathcal{Z}_{n})$, with offspring probabilities $(p_{i})$ and immigration probabilities $(q_{i})$. In the case when $p_{0}=0$, $p_{1}\neq0$, $q_{0}=0$ (that is, when $\operatorname{essinf}(\mathcal{Z}_{n})$ grows linearly in $n$), we establish the asymptotics of the left tail $\mathbb{P}\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow0$, of the martingale limit $\mathcal{W}$ of the process $(\mathcal{Z}_{n})$. Further, we consider the first generation $\mathcal{K}$ such that $\mathcal{Z}_{\mathcal{K}}\operatorname{essinf}(\mathcal{Z}_{\mathcal{K}})$ and study the asymptotic behaviour of $\mathcal{K}$ conditionally on $\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow 0$. We find the growth scale and the fluctuations of $\mathcal{K}$ and compare the results with those for standard Galton–Watson processes.

14. #### Statistical inference for the doubly stochastic self-exciting process

Clinet, Simon; Potiron, Yoann
We introduce and show the existence of a Hawkes self-exciting point process with exponentially-decreasing kernel and where parameters are time-varying. The quantity of interest is defined as the integrated parameter $T^{-1}\int_{0}^{T}\theta_{t}^{*}\,dt$, where $\theta_{t}^{*}$ is the time-varying parameter, and we consider the high-frequency asymptotics. To estimate it naïvely, we chop the data into several blocks, compute the maximum likelihood estimator (MLE) on each block, and take the average of the local estimates. The asymptotic bias explodes asymptotically, thus we provide a non-naïve estimator which is constructed as the naïve one when applying a first-order bias reduction to the local MLE. We...

15. #### A unified matrix model including both CCA and F matrices in multivariate analysis: The largest eigenvalue and its applications

Han, Xiao; Pan, Guangming; Yang, Qing
Let $\mathbf{Z}_{M_{1}\times N}=\mathbf{T}^{\frac{1}{2}}\mathbf{X}$ where $(\mathbf{T}^{\frac{1}{2}})^{2}=\mathbf{T}$ is a positive definite matrix and $\mathbf{X}$ consists of independent random variables with mean zero and variance one. This paper proposes a unified matrix model $\mathbf{\Omega}=(\mathbf{Z}\mathbf{U}_{2}\mathbf{U}_{2}^{T}\mathbf{Z}^{T})^{-1}\mathbf{Z}\mathbf{U}_{1}\mathbf{U}_{1}^{T}\mathbf{Z}^{T},$ where $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are isometric with dimensions $N\times N_{1}$ and $N\times(N-N_{2})$ respectively such that $\mathbf{U}_{1}^{T}\mathbf{U}_{1}=\mathbf{I}_{N_{1}}$, $\mathbf{U}_{2}^{T}\mathbf{U}_{2}=\mathbf{I}_{N-N_{2}}$ and $\mathbf{U}_{1}^{T}\mathbf{U}_{2}=0$. Moreover, $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ (random or non-random) are independent of $\mathbf{Z}_{M_{1}\times N}$ and with probability tending to one, $\operatorname{rank}(\mathbf{U}_{1})=N_{1}$ and $\operatorname{rank}(\mathbf{U}_{2})=N-N_{2}$. We establish the asymptotic Tracy–Widom distribution for its largest eigenvalue under moment assumptions on $\mathbf{X}$ when $N_{1},N_{2}$ and $M_{1}$ are comparable. ¶ The asymptotic distributions of the maximum eigenvalues of...

16. #### Expected number and height distribution of critical points of smooth isotropic Gaussian random fields

Cheng, Dan; Schwartzman, Armin
We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae depend on the covariance function...

17. #### The Gamma Stein equation and noncentral de Jong theorems

Döbler, Christian; Peccati, Giovanni
We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate $U$-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong (J. Multivariate Anal. 34 (1990) 275–289), Nourdin, Peccati and Reinert (Ann. Probab. 38 (2010) 1947–1985) and Döbler and Peccati (Electron. J. Probab. 22 (2017) no. 2), (ii)...

18. #### Parametric inference for nonsynchronously observed diffusion processes in the presence of market microstructure noise

Ogihara, Teppei
We study parametric inference for diffusion processes when observations occur nonsynchronously and are contaminated by market microstructure noise. We construct a quasi-likelihood function and study asymptotic mixed normality of maximum-likelihood- and Bayes-type estimators based on it. We also prove the local asymptotic normality of the model and asymptotic efficiency of our estimator when the diffusion coefficients are deterministic and noise follows a normal distribution. We conjecture that our estimator is asymptotically efficient even when the latent process is a general diffusion process. An estimator for the quadratic covariation of the latent process is also constructed. Some numerical examples show that...

19. #### Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models

Bartroff, Jay; Goldstein, Larry; Işlak, Ümit
Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results for random graphs, germ-grain models in stochastic geometry and multinomial allocation models. The results obtained compare favorably with classical methods, including the use of McDiarmid’s inequality, negative association, and self bounding functions.

20. #### Entropy production in nonlinear recombination models

Caputo, Pietro; Sinclair, Alistair
We study the convergence to equilibrium of a class of nonlinear recombination models. In analogy with Boltzmann’s H-theorem from kinetic theory, and in contrast with previous analysis of these models, convergence is measured in terms of relative entropy. The problem is formulated within a general framework that we refer to as Reversible Quadratic Systems. Our main result is a tight quantitative estimate for the entropy production functional. Along the way, we establish some new entropy inequalities generalizing Shearer’s and related inequalities.

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