## Recursos de colección

Bernoulli

1. #### On branching process with rare neutral mutation

Blancas, Airam; Rivero, Víctor
In this paper, we study the genealogical structure of a Galton–Watson process with neutral mutations. Namely, we extend in two directions the asymptotic results obtained in Bertoin [Stochastic Process. Appl. 120 (2010) 678–697]. In the critical case, we construct the version of the model in Bertoin [Stochastic Process. Appl. 120 (2010) 678–697], conditioned not to be extinct. We establish a version of the limit theorems in Bertoin [Stochastic Process. Appl. 120 (2010) 678–697], when the reproduction law has an infinite variance and it is in the domain of attraction of an $\alpha$-stable distribution, both for the unconditioned process and for...

2. #### Asymptotic analysis of covariance parameter estimation for Gaussian processes in the misspecified case

Bachoc, François
In parametric estimation of covariance function of Gaussian processes, it is often the case that the true covariance function does not belong to the parametric set used for estimation. This situation is called the misspecified case. In this case, it has been shown that, for irregular spatial sampling of observation points, Cross Validation can yield smaller prediction errors than Maximum Likelihood. Motivated by this observation, we provide a general asymptotic analysis of the misspecified case, for independent and uniformly distributed observation points. We prove that the Maximum Likelihood estimator asymptotically minimizes a Kullback–Leibler divergence, within the misspecified parametric set, while...

3. #### Exact and fast simulation of max-stable processes on a compact set using the normalized spectral representation

Oesting, Marco; Schlather, Martin; Zhou, Chen
The efficiency of simulation algorithms for max-stable processes relies on the choice of the spectral representation: different choices result in different sequences of finite approximations to the process. We propose a constructive approach yielding a normalized spectral representation that solves an optimization problem related to the efficiency of simulating max-stable processes. The simulation algorithm based on the normalized spectral representation can be regarded as max-importance sampling. Compared to other simulation algorithms hitherto, our approach has at least two advantages. First, it allows the exact simulation of a comprising class of max-stable processes. Second, the algorithm has a stopping time with...

4. #### Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions

Kusuoka, Seiichiro; Tudor, Ciprian A.
We give necessary and sufficient conditions to characterize the convergence in distribution of a sequence of arbitrary random variables to a probability distribution which is the invariant measure of a diffusion process. This class of target distributions includes the most known continuous probability distributions. Precisely speaking, we characterize the convergence in total variation to target distributions which are not Gaussian or Gamma distributed, in terms of the Malliavin calculus and of the coefficients of the associated diffusion process. We also prove that, among the distributions whose associated squared diffusion coefficient is a polynomial of second degree (with some restrictions on...

5. #### Maximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time series

Bücher, Axel; Segers, Johan
The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution to a sample of block maxima extracted from an observed stretch of a time series. The method is usually validated under two simplifying assumptions: the block maxima should be distributed exactly according to an extreme-value distribution and the sample of block maxima should be independent. Both assumptions are only approximately true. The present paper validates that the simplifying assumptions can in fact be safely made. ¶ For general triangular arrays of block maxima attracted to the Fréchet distribution, consistency and asymptotic normality is established for the maximum likelihood estimator...

6. #### American options with asymmetric information and reflected BSDE

Esmaeeli, Neda; Imkeller, Peter
We consider an American contingent claim on a financial market where the buyer has additional information. Both agents (seller and buyer) observe the same prices, while the information available to them may differ due to some extra exogenous knowledge the buyer has. The buyer’s information flow is modeled by an initial enlargement of the reference filtration. It seems natural to investigate the value of the American contingent claim with asymmetric information. We provide a representation for the cost of the additional information relying on some results on reflected backward stochastic differential equations (RBSDE). This is done by using an interpretation...

7. #### The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model

Janssen, Anja; Mikosch, Thomas; Rezapour, Mohsen; Xie, Xiaolei
We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $\alpha$-stable limit distributions of the sample covariance matrix. For the case of heavy-tailed innovations, we show that...

8. #### Quenched invariance principles for the discrete Fourier transforms of a stationary process

Barrera, David
In this paper, we study the asymptotic behavior of the normalized cadlag functions generated by the discrete Fourier transforms of a stationary centered square-integrable process, started at a point. ¶ We prove that the quenched invariance principle holds for averaged frequencies under no assumption other than ergodicity, and that this result holds also for almost every fixed frequency under a certain generalization of the Hannan condition and a certain rotated form of the Maxwell and Woodroofe condition which, under a condition of weak dependence that we specify, is guaranteed for a.e. frequency. If the process is in particular weakly mixing, our results...

9. #### Bump detection in heterogeneous Gaussian regression

Enikeeva, Farida; Munk, Axel; Werner, Frank
We analyze the effect of a heterogeneous variance on bump detection in a Gaussian regression model. To this end, we allow for a simultaneous bump in the variance and specify its impact on the difficulty to detect the null signal against a single bump with known signal strength. This is done by calculating lower and upper bounds, both based on the likelihood ratio. ¶ Lower and upper bounds together lead to explicit characterizations of the detection boundary in several subregimes depending on the asymptotic behavior of the bump heights in mean and variance. In particular, we explicitly identify those regimes, where the...

10. #### Smooth backfitting for additive modeling with small errors-in-variables, with an application to additive functional regression for multiple predictor functions

Han, Kyunghee; Müller, Hans-Georg; Park, Byeong U.
We study smooth backfitting when there are errors-in-variables, which is motivated by functional additive models for a functional regression model with a scalar response and multiple functional predictors that are additive in the functional principal components of the predictor processes. The development of a new smooth backfitting technique for the estimation of the additive component functions in functional additive models with multiple functional predictors requires to address the difficulty that the eigenfunctions and therefore the functional principal components of the predictor processes, which are the arguments of the proposed additive model, are unknown and need to be estimated from the...

11. #### Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes

Inoue, Akihiko; Kasahara, Yukio; Pourahmadi, Mohsen
For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the finite predictor coefficients, prediction error covariances, and the PACF as well as for proving...

12. #### Determinantal point process models on the sphere

Møller, Jesper; Nielsen, Morten; Porcu, Emilio; Rubak, Ege
We consider determinantal point processes on the $d$-dimensional unit sphere $\mathbb{S}^{d}$. These are finite point processes exhibiting repulsiveness and with moment properties determined by a certain determinant whose entries are specified by a so-called kernel which we assume is a complex covariance function defined on $\mathbb{S}^{d}\times\mathbb{S}^{d}$. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. Particularly, we characterize and construct isotropic DPPs models on $\mathbb{S}^{d}$, where it becomes essential to specify the eigenvalues and eigenfunctions in a spectral representation for the kernel, and we figure out how repulsive isotropic DPPs can...

13. #### Critical points of multidimensional random Fourier series: Central limits

Nicolaescu, Liviu I.
We investigate certain families $X^{\hbar}$, $0<\hbar\ll1$ of stationary Gaussian random smooth functions on the $m$-dimensional torus $\mathbb{T}^{m}:=\mathbb{R}^{m}/\mathbb{Z}^{m}$ approaching the white noise as $\hbar\to0$. We show that there exists universal constants $c_{1},c_{2}>0$ such that for any cube $B\subset\mathbb{R}^{m}$ of size $r\leq1/2$, the number of critical points of $X^{\hbar}$ in the region $B\bmod\mathbb{Z}^{m}\subset\mathbb{T}^{m}$ has mean $\sim c_{1}\operatorname{vol}(B)\hbar^{-m}$, variance $\sim c_{2}\operatorname{vol}(B)\hbar^{-m}$, and satisfies a central limit theorem as $\hbar\searrow0$.

14. #### Efficient estimation for generalized partially linear single-index models

Wang, Li; Cao, Guanqun
In this paper, we study the estimation for generalized partially linear single-index models, where the systematic component in the model has a flexible semi-parametric form with a general link function. We propose an efficient and practical approach to estimate the single-index link function, single-index coefficients as well as the coefficients in the linear component of the model. The estimation procedure is developed by applying quasi-likelihood and polynomial spline smoothing. We derive large sample properties of the estimators and show the convergence rate of each component of the model. Asymptotic normality and semiparametric efficiency are established for the coefficients in both...

15. #### On matrix estimation under monotonicity constraints

Chatterjee, Sabyasachi; Guntuboyina, Adityanand; Sen, Bodhisattva
We consider the problem of estimating an unknown $n_{1}\times n_{2}$ matrix $\mathbf{\theta}^{*}$ from noisy observations under the constraint that $\mathbf{\theta}^{*}$ is nondecreasing in both rows and columns. We consider the least squares estimator (LSE) in this setting and study its risk properties. We show that the worst case risk of the LSE is $n^{-1/2}$, up to multiplicative logarithmic factors, where $n=n_{1}n_{2}$ and that the LSE is minimax rate optimal (up to logarithmic factors). We further prove that for some special $\mathbf{\theta}^{*}$, the risk of the LSE could be much smaller than $n^{-1/2}$; in fact, it could even be parametric, that...

16. #### Inference for a two-component mixture of symmetric distributions under log-concavity

In this article, we revisit the problem of estimating the unknown zero-symmetric distribution in a two-component location mixture model, considered in previous works, now under the assumption that the zero-symmetric distribution has a log-concave density. When consistent estimators for the shift locations and mixing probability are used, we show that the nonparametric log-concave Maximum Likelihood estimator (MLE) of both the mixed density and that of the unknown zero-symmetric component are consistent in the Hellinger distance. In case the estimators for the shift locations and mixing probability are $\sqrt{n}$-consistent, we establish that these MLE’s converge to the truth at the rate...

17. #### Functional central limit theorems in $L^{2}(0,1)$ for logarithmic combinatorial assemblies

Tsukuda, Koji
Functional central limit theorems in $L^{2}(0,1)$ for logarithmic combinatorial assemblies are presented. The random elements argued in this paper are viewed as elements taking values in $L^{2}(0,1)$ whereas the Skorokhod space is argued as a framework of weak convergences in functional central limit theorems for random combinatorial structures in the literature. It enables us to treat other standardized random processes which converge weakly to a corresponding Gaussian process with additional assumptions.

18. #### Exponential mixing properties for time inhomogeneous diffusion processes with killing

Del Moral, Pierre; Villemonais, Denis
We consider an elliptic and time-inhomogeneous diffusion process with time-periodic coefficients evolving in a bounded domain of $\mathbb{R}^{d}$ with a smooth boundary. The process is killed when it hits the boundary of the domain (hard killing) or after an exponential time (soft killing) associated with some bounded rate function. The branching particle interpretation of the non absorbed diffusion again behaves as a set of interacting particles evolving in an absorbing medium. Between absorption times, the particles evolve independently one from each other according to the diffusion evolution operator; when a particle is absorbed, another selected particle splits into two offsprings....

19. #### Mixing time and cutoff for a random walk on the ring of integers mod $n$

Bate, Michael; Connor, Stephen
We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive ‘step’ or a multiplicative ‘jump’. When the probability of making a jump tends to zero as an appropriate power of $n$, we prove the existence of a total variation pre-cutoff for this walk. In addition, we show that the process obtained by subsampling our walk at jump times exhibits a true cutoff, with mixing time dependent on whether the step distribution has zero mean.

20. #### An upper bound on the convergence rate of a second functional in optimal sequence alignment

Hauser, Raphael; Matzinger, Heinrich; Popescu, Ionel
Consider finite sequences $X_{[1,n]}=X_{1},\ldots,X_{n}$ and $Y_{[1,n]}=Y_{1},\ldots,Y_{n}$ of length $n$, consisting of i.i.d. samples of random letters from a finite alphabet, and let $S$ and $T$ be chosen i.i.d. randomly from the unit ball in the space of symmetric scoring functions over this alphabet augmented by a gap symbol. We prove a probabilistic upper bound of linear order in $(\ln(n))^{1/4}n^{3/4}$ for the deviation of the score relative to $T$ of optimal alignments with gaps of $X_{[1,n]}$ and $Y_{[1,n]}$ relative to $S$. It remains an open problem to prove a lower bound. Our result contributes to the understanding of the microstructure of...

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