This paper is a complement to the studies on the minimum of a real-valued branching random walk. In the boundary case [Electron. J. Probab. 10 (2005) 609–631], Aïdékon in a seminal paper [Ann. Probab. 41 (2013) 1362–1426] obtained the convergence in law of the minimum after a suitable renormalization. We study here the situation when the log-generating function of the branching random walk explodes at some positive point and it cannot be reduced to the boundary case. In the associated thermodynamics framework, this corresponds to a first-order phase transition, while the boundary case corresponds to a second-order phase transition.

We establish quantitative bounds for rates of convergence and asymptotic variances for iterated conditional sequential Monte Carlo (i-cSMC) Markov chains and associated particle Gibbs samplers [J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 (2010) 269–342]. Our main findings are that the essential boundedness of potential functions associated with the i-cSMC algorithm provide necessary and sufficient conditions for the uniform ergodicity of the i-cSMC Markov chain, as well as quantitative bounds on its (uniformly geometric) rate of convergence. Furthermore, we show that the i-cSMC Markov chain cannot even be geometrically ergodic if this essential boundedness does not hold in many applications of interest. Our sufficiency and quantitative bounds rely on a novel non-asymptotic analysis of the expectation of a standard normalizing constant estimate with respect to a “doubly conditional” SMC algorithm. In addition, our results for i-cSMC imply that the rate of convergence can be improved arbitrarily by increasing $N$, the number of particles in the algorithm, and that in the presence of mixing assumptions, the rate of convergence can be kept constant by increasing $N$ linearly with the time horizon. We translate the sufficiency of the boundedness condition for i-cSMC into sufficient conditions for the particle Gibbs Markov chain to be geometrically ergodic and quantitative bounds on its geometric rate of convergence, which imply convergence of properties of the particle Gibbs Markov chain to those of its corresponding Gibbs sampler. These results complement recently discovered, and related, conditions for the particle marginal Metropolis–Hastings (PMMH) Markov chain.

For each probability distribution on a countable alphabet, a sequence of positive functionals are developed as tail indices. By and only by the asymptotic behavior of these indices, domains of attraction for all probability distributions on the alphabet are defined. The three main domains of attraction are shown to contain distributions with thick tails, thin tails and no tails respectively, resembling in parallel the three main domains of attraction, Fréchet, Gumbel and Weibull families, for continuous random variables on the real line. In addition to the probabilistic merits associated with the domains, the tail indices are partially motivated by the fact that there exists an unbiased estimator for every index in the sequence, which is therefore statistically observable, provided that the sample is sufficiently large.

Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. Instead we consider the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the eigenvectors under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.

Inference for the parametric distribution of a response given covariates is considered under informative selection of a sample from a finite population. Under this selection, the conditional distribution of a response in the sample, given the covariates and given that it was selected for observation, is not the same as the conditional distribution of the response in the finite population, given only the covariates. It is instead a weighted version of the conditional distribution of interest. Inference must be modified to account for this informative selection. An established approach in this context is maximum “sample likelihood”, developing a weight function that reflects the informative sampling design, then treating the observations as if they were independently distributed according to the weighted distribution. While the sample likelihood methodology has been widely applied, its theoretical foundation has been less developed. A precise asymptotic description of a wide range of informative selection mechanisms is proposed. Under this framework, consistency and asymptotic normality of the maximum sample likelihood estimators are established. The theory allows for the possibility of nuisance parameters that describe the selection mechanism. The proposed regularity conditions are verifiable for various sample schemes, motivated by real problems in surveys. Simulation results for these examples illustrate the quality of the asymptotic approximations, and demonstrate a practical approach to variance estimation that combines aspects of model-based information theory and design-based variance estimation.

A Hörmander-type theorem is established for Itô processes and related backward stochastic partial differential equations (BSPDEs). A short self-contained proof is also provided for the $L^{2}$-theory of linear, possibly degenerate BSPDEs, in which new gradient estimates are obtained.

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 optimal alignments relative to one given scoring function, extending a theory begun in (J. Stat. Phys. 153 (2013) 512–529).

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.

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. This article is concerned with the stability properties of these non absorbed processes. Under some classical ellipticity properties on the diffusion process and some mild regularity properties of the hard obstacle boundaries, we prove an uniform exponential strong mixing property of the process conditioned to not be killed. We also provide uniform estimates w.r.t. the time horizon for the interacting particle interpretation of these non-absorbed processes, yielding what seems to be the first result of this type for this class of diffusion processes evolving in soft and hard obstacles, both in homogeneous and non-homogeneous time settings.

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.

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 $n^{-2/5}$ in the $L_{1}$ distance. To estimate the shift locations and mixing probability, we use the estimators proposed by (Ann. Statist. 35 (2007) 224–251). The unknown zero-symmetric density is efficiently computed using the R package logcondens.mode.

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 is, $n^{-1}$ up to logarithmic factors. Such parametric rates occur when the number of “rectangular” blocks of $\mathbf{\theta}^{*}$ is bounded from above by a constant. We also derive an interesting adaptation property of the LSE which we term variable adaptation – the LSE adapts to the “intrinsic dimension” of the problem and performs as well as the oracle estimator when estimating a matrix that is constant along each row/column. Our proofs, which borrow ideas from empirical process theory, approximation theory and convex geometry, are of independent interest.

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 the single-index and linear components. By making use of spline basis approximation and Fisher score iteration, our approach has numerical advantages in terms of computing efficiency and stability in practice. Both simulated and real data examples are used to illustrate our proposed methodology.

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 $\hbar\searrow0$.

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 be. Moreover, we discuss the shortcomings of adapting existing models for isotropic covariance functions and consider strategies for developing new models, including a useful spectral approach.

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 a multivariate version of Baxter’s inequality for a multivariate FARIMA process with a common fractional differencing order for all components of the process.

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 data. The available estimated functional principal components contain an error that is small for large samples but nevertheless affects the estimation of the additive component functions. This error-in-variables situation requires to develop new asymptotic theory for smooth backfitting. Our analysis also pertains to general situations where one encounters errors in the predictors for an additive model, when the errors become smaller asymptotically. We also study the finite sample properties of the proposed method for the application in functional additive regression through a simulation study and a real data example.

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 additional information about a simultaneous bump in variance eases the detection problem for the signal. This effect is made explicit in the constant and/or the rate, appearing in the detection boundary.

We also discuss the case of an unknown bump height and provide an adaptive test and some upper bounds in that case.

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 describe the asymptotic distributions of the normalized discrete Fourier transforms at every frequency other than $0$ and $\pi$ under the generalized Hannan condition.

We prove also that under a certain regularity hypothesis the conditional centering is irrelevant for averaged frequencies, and that the same holds for a given fixed frequency under the rotated Maxwell and Woodroofe condition but not necessarily under the generalized Hannan condition. In particular, this implies that the hypothesis of regularity is not sufficient for functional convergence without random centering at a.e. fixed frequency.

The proofs are based on martingale approximations and combine results from Ergodic theory of recent and classical origin with approximation results by contemporary authors and with some facts from Harmonic Analysis and Functional Analysis.

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 the limiting behavior resembles that of completely independent observations. In contrast to this, for a heavy-tailed volatility sequence the possible limiting behavior is more diverse and allows for dependencies in the limiting distributions which are determined by the structure of the underlying volatility sequence.

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 of prices of American contingent claims with extra information for the buyer by solutions of appropriate RBSDE.

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 of the parameters of the limiting Fréchet distribution. The results are specialized to the common setting of block maxima extracted from a strictly stationary time series. The case where the underlying random variables are independent and identically distributed is further worked out in detail. The results are illustrated by theoretical examples and Monte Carlo simulations.

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 its coefficients), the only possible limits of sequences of multiple integrals are the Gaussian and the Gamma laws.

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 finite expectation. In practice, our approach has the potential of considerably reducing the simulation time of max-stable processes.

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 Cross Validation asymptotically minimizes the integrated square prediction error. In Monte Carlo simulations, we show that the covariance parameters estimated by Maximum Likelihood and Cross Validation, and the corresponding Kullback–Leibler divergences and integrated square prediction errors, can be strongly contrasting. On a more technical level, we provide new increasing-domain asymptotic results for independent and uniformly distributed observation points.

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 the process conditioned to nonextinction. In the latter case, we obtain the convergence (after re-normalization) of the allelic sub-populations towards a tree indexed CSBP with immigration.