在适当条件下，我证明了向量值平稳随机场非线性泛函的非中心极限定理。它们是文献\cite{2}中结果的多变量版本。此前，A. M. Arcones 在文献\cite{1}中提出了一个定理，可视为这些结果的多变量推广。但我发现 Arcones 的讨论并不完整，我认为要给出完整的证明，首先必须建立向量值高斯平稳随机场理论的更深厚基础。这项工作已在我的文献\cite{4}中完成，使我能够将文献\cite{2}中的方法应用于向量值情形的研究。在此，借助该基础，我证明了文献\cite{2}中结果的多变量版本。
英文摘要：
Here I prove non-central limit theorems for non-linear functionals of vector valued stationary random fields under appropriate conditions. They are the multivariate versions of the results in paper\cite{2}. Previously A. M. Arcones formulated a theorem in paper\cite{1} which can be considered as the multivariate generalization of these results. But I found Arcones' discussion incomplete, and in my opinion to give a complete proof first a more profound foundation of the theory of vector valued Gaussian stationary random fields has to be worked out. This was done in my paper\cite{4} which enabled me to adapt the method in paper\cite{2} to the study of the vector valued case. Here I prove with its help the desired multivariate version of the results in paper\cite{2}.
We prove a complete class theorem that characterizes \emph{all} stationary time reversible Markov processes whose finite dimensional marginal distributions (of all orders) are infinitely divisible. Aside from two degenerate cases (iid and constant), in both discrete and continuous time every such process with full support is a branching process with Poisson or Negative Binomial marginal univariate distributions and a specific bivariate distribution at pairs of times. As a corollary, we prove that every nondegenerate stationary integer valued processes constructed by the Markov thinning process…
The seminal papers of Pickands [1,2] paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian's Lemma, Borell-TIS and Piterbarg inequalities there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting obtaining the exact asymptotics of the exceedance probabilities for both stationary…
We give a pathwise construction of a two-parameter family of purely-atomic-measure-valued diffusions in which ranked masses of atoms are stationary with the Poisson-Dirichlet$(α,θ)$ distributions, for $α\in (0,1)$ and $θ\ge 0$. This resolves a conjecture of Feng and Sun (2010). We build on our previous work on $(α,0)$- and $(α,α)$-interval partition evolutions. Indeed, we first extract a self-similar superprocess from the levels of stable processes whose jumps are decorated with squared Bessel excursions and distinct allelic types. We complete our construction by time-change and normalisation …
In this paper we present a stability criterion for finite measure-valued stochastic recursions, generalizing Loynes's Theorem to spaces of measures. This result provides conditions for the reach of a "total stationary state" for the queue with an infinity of servers and the single-server SRPT queue. Indeed, we give in both cases a condition of existence of a stationary measure-valued recursive sequence characterizing the queueing system exhaustively.
In this paper we study the asymptotic normality of the normalized partial sum of a Hilbert-space valued strictly stationary random field satisfying the interlaced $ρ'$-mixing condition.
In this paper the question of the extent to which truncated heavy tailed random vectors, taking values in a Banach space, retain the characteristic features of heavy tailed random vectors, is answered from the point of view of the central limit theorem.
For $M \geq 2$ we construct a strictly stationary, $M$-tuplewise independent sequence that is mixing (in the ergodic-theoretic sense) and yet still fails to satisfy the Central Limit Theorem.
We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of a strictly stationary orthomartingale random field. These inequalities can be used to establish complete convergence of normalized maxima of partial sums.
In this paper, we formulate and prove new properties of conditional quantiles given one of the particular sigma-fields. Next, we use them to investigate almost sure asymptotic behavior of central order statistics which arise from strictly stationary processes. Specifically we provide a new version of a strong ergodic theorem for central order statistics.
We find a sufficient condition under which a central limit theorem for a stationary linear process is quenched. We find a stationary linear process szatisfying the Maxwell-Woodroofe condition for which the variances of partial sums are o(n), there is a CLT with a convergence towards N(0,1) when dividing by standard deviation of the partial sums, and the CLT is not quenched. The weak invariance principle does not hold.
In this paper we show that any one-dimensional stationary, finite-valued Markov Random Field (MRF) is a Markov chain, without any mixing condition or condition on the support. Our proof makes use of two properties of the support $X$ of a finite-valued stationary MRF: 1) $X$ is non-wandering (this is a property of the support of any finite-valued stationary process) and 2) $X$ is a topological Markov field (TMF). The latter is a new property that sits in between the classes of shifts of finite type and sofic shifts, which are well-known objects of study in symbolic dynamics. Here, we develop th…
We investigate the connection between conditional local limit theorems and the local time of integer-valued stationary processes. We show that a conditional local limit theorem (at 0) implies the convergence of local times to Mittag-Leffler distributions, both in the weak topology of distributions and a.s. in the space of distributions.
In this paper we extend a central limit theorem of Peligrad for uniformly strong mixing random fields satisfying the Lindeberg condition in the absence of stationarity property. More precisely, we study the asymptotic normality of the partial sums of uniformly $α$-mixing non-stationary random fields satisfying the Lindeberg condition, in the presence of an extra dependence assumption involving maximal correlations.
Consider a stationary, linear Hilbert space valued process. We establish Berry-Essen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators. The case of non-linear Bernoulli-shift sequences is also considered. If the sequence is $m$-dependent, the optimal rate $(n/m)^{1/2}$ is reached. If the sequence is weakly geometrically dependent, the rate $(n/\log n)^{1/2}$ is obtained.
We prove a functional central limit theorem for integrals $\int_W f(X(t))\, dt$, where $(X(t))_{t\in\mathbb{R}^d}$ is a stationary mixing random field and the stochastic process is indexed by the function $f$, as the integration domain $W$ grows in Van Hove-sense. We discuss properties of the covariance function of the asymptotic Gaussian process.
The first aim of this paper is to wonder to what extent we can generalize the central limit theorem of Gordin [5] under the so-called L 1-projective criteria to ergodic stationary random fields when completely commuting filtrations are considered. Surprisingly it appears that this result cannot be extended to its full generality and that an additional condition is needed.
In 1983, N. Herrndorf proved that for a $ϕ$-mixing sequence satisfying the central limit theorem and $\liminf_{n\to\infty}\frac{σ^2_n}n>0$, the weak invariance principle takes place. The question whether for strictly stationary sequences with finite second moments and a weaker type ($α$, $β$, $ρ$) of mixing the central limit theorem implies the weak invariance principle remained open. We construct a strictly stationary $β$-mixing sequence with finite moments of any order and linear variance for which the central limit theorem takes place but not the weak invariance principle.
Let $\boldsymbol{X}(t)=(X_1(t),\ldots,X_d(t))$ be a Gaussian vector process and $g(t)$ be a continuous function. The asymptotics of distribution of $\left\|\boldsymbol{X}(t)\right\|_p$, the $L^p$ norm for Gaussian finite-dimensional vector, have been investigated in numerous literatures. In this contribution we are concerned with the exact tail asymptotics of $\left\|\boldsymbol{X}(t)\right\|^c_p,\ c>0, $ with trend $g(t)$ over $[0,T]$. Both scenarios that $\boldsymbol{X}(t)$ is locally stationary and non-stationary are considered. Important examples include $\sum_{i=1}^d \left|X_i(t)\right|+g…
Motivated by a random matrix theory model from wireless communications, we define random operator-valued matrices as the elements of $L^{\infty-}(Ω,{\mathcal F},{\mathbb P}) \otimes M_d({\mathcal A})$ where $(Ω,{\mathcal F},{\mathbb P})$ is a classical probability space and $({\mathcal A},φ)$ is a non-commutative probability space. A central limit theorem for the mean $M_d(\mathbb{C})$-valued moments of these random operator-valued matrices is derived. Also a numerical algorithm to compute the mean $M_d({\mathbb C})$-valued Cauchy transform of operator-valued semicircular mixtures is analyzed.
We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fréchet means) of independent non-identically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg-Feller central approximation theorem for vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem for…
We derive in this preprint the moment and exponential tail estimates, sufficient conditions for the Non-Central Limit Theorem (NCLT) in the ordinary one-dimensional space as well as in the space of continuous functions for the properly (natural) normalized multi-indexed sums of function of random variables, processes or fields (r.f.), on the other words V-statistics, parametric, in general case. We construct also some examples in order to show the exactness of obtained estimates. We will use the theory of the so-called degenerate approximation of the functions of several variables as well as t…
We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only positive dependence could be treated. The negative dependence involves cancellations of the Gaussian second order. This leads to new types of limiting processes involving stable random measures, due to heavy tails, Mittag-Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations.
We consider a $\mathbb{R}^d$-valued branching random walk with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$. With the help of the uniform convergence of martingale and the multifractal analysis, we establish a large deviation result for the measures $Z_n$ as well as a moderate deviation principle.
We consider asymptotic behavior of Fourier transforms of stationary ergodic sequences with finite second moments. We establish a central limit theorem (CLT) for almost all frequencies and also an annealed CLT. The theorems hold for all regular sequences. Our results shed new light on the foundation of spectral analysis and on the asymptotic distribution of periodogram, and it provides a nice blend of harmonic analysis, theory of stationary processes and theory of martingales.
A strictly stationary sequence of random variables is constructed with the following properties: (i) the random variables take the values -1 and +1 with probability 1/2 each, (ii) every five of the random variables are independent, (iii) the sequence is "causal" in a certain sense, (iv) the sequence has a trivial double tail sigma-field, and (v) regardless of the normalization used, the partial sums do not converge to a (nondegenerate) normal law. The example has some features in common with a recent construction (for an arbitrary fixed positive integer N), by Alexander Pruss and the author, o…
General Central limit theorem deals with weak limits (in type) of sums of row-elements of array random variables. In some situations as in the invariance principle problem, the sums may include only parts of the row-elements. For strictly stationary arrays (stationary for each row), there is no change to the asymptotic results. But for non-stationary data, especially for dependent data, asymptotic laws of partial sums moving in rows may require extra-conditions to exist. This paper deals with central limit theorems with Gaussian limits for non-stationary data. Our main focus is on dependent da…
Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard
我们展示如何里约的方法[概率。Theory Related Fields 104（1996）255- 282]可以适用于在Kantorovich度量的意义下建立某些平稳过程的多维中心极限定理中${\frac{1}{\sqrt{n}}}$的收敛速率。我们给出这个一般结果的两个应用：在克努森气体的情况下，在西奈台球的情况下。
We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying Lévy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments that coincides on a part of its parameter space with a previously described process. The normalizing sequence and the limiting process are determined by the ergodic-theoretical properties of the flow underlying the integral representation of the process. These properties can be interpreted as determining how long the…
We study the probability of a real-valued stationary process to be positive on a large interval $[0,N]$. We show that if in some neighborhood of the origin the spectral measure of the process has density which is bounded away from zero and infinity, then the decay of this probability is bounded between two exponential functions in $N$. This generalizes similar bounds obtained for particular cases, such as a recent result by Artezana, Buckley, Marzo, Olsen.
\noindent The paper establishes weak convergence in $C[0,1]$ of normalized stochastic processes, generated by Toeplitz type quadratic functionals of a continuous time Gaussian stationary process, exhibiting long-range dependence. Both central and non-central functional limit theorems are obtained.
We prove that, for (adapted) stationary processes, the so-called Maxwell-Wood-roofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. We obtain a similar conclusion concerning the Marcinkiewicz-zygmund strong law of large numbers. Those results actually hold in the context of Banach valued stationary processes, including the case of $L^r$-valued random variables, with $1\le r<\infty$. In this setting we also prove the weak invariance principle, under a version of the Maxwell-Woodroofe condition, generalizing a result of Peligrad and Utev \cite{…
Our main result is to prove almost-sure convergence of a stochastic-approximation algorithm defined on the space of measures on a non-compact space. Our motivation is to apply this result to measure-valued Pólya processes (MVPPs, also known as infinitely-many Pólya urns). Our main idea is to use Foster-Lyapunov type criteria in a novel way to generalize stochastic-approximation methods to measure-valued Markov processes with a non-compact underlying space, overcoming in a fairly general context one of the major difficulties of existing studies on this subject. From the MVPPs point of view, our…
Let $(X_{k})_{k \in \mathbb Z }$ be a linear process with values in a separable Hilbert space $\mathbb{H}$ given by $X_{k} =\sum_{j=0}^{\infty} (j+1)^{-N}\varepsilon_{k-j}$ for each $k \in \mathbb Z$, where $N:\mathbb{H} \to \mathbb{H}$ is a bounded, linear normal operator and $(\varepsilon_{k})_{ k \in \mathbb Z }$ is a sequence of independent, identically distributed $\mathbb{H}$-valued random variables with $E\varepsilon_{0}=0$ and $E\| \varepsilon_{0} \|^2<\infty$. We investigate the central and the functional central limit theorem for $(X_{k})_{k \in \mathbb Z }$ when the series of operat…
Max-stable random fields are very appropriate for the statistical modelling of spatial extremes. Hence, integrals of functions of max-stable random fields over a given region can play a key role in the assessment of the risk of natural disasters, meaning that it is relevant to improve our understanding of their probabilistic behaviour. For this purpose, in this paper, we propose a general central limit theorem for functions of stationary max-stable random fields on $\mathbb{R}^d$. Then, we show that appropriate functions of the Brown-Resnick random field with a power variogram and of the Smith…
In this note we (in particular) prove an almost sure invariance principle (ASIP) for non-stationary and uniformly bounded sequences of random variables which are exponentially fast $ϕ$-mixing. The obtained rate is of order $o(V_n^{\frac14+\del})$ for an arbitrary $\del>0$, where $V_n$ is the variance of the underlying partial sums $S_n$. For certain classes of inhomogeneous Markov chains we also prove a vector-valued ASIP with similar rates.
Let $X=\{X_t: t\ge 0\}$ be a stationary piecewise continuous $\R^d$-valued process that moves between jumps along the integral curves of a given continuous vector field, and let $S\subset\R^d$ be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of $S$ by $X$ to the distribution of $X_0$. Our result is illustrated by examples relating to queueing networks and stress release network models.
In this paper, we give rates of convergence, for minimal distances and for the uniform distance, between the law of partial sums of martingale differences and thelimiting Gaussian distribution. More precisely, denoting by $P_{X}$ the law of a random variable $X$ and by $G_{a}$ the normal distribution ${\mathcal N} (0,a)$, we are interested by giving quantitative estimates for the convergence of $P_{S_n/\sqrt{V_n}}$ to $G_1$, where $S_n$ is the partial sum associated with either martingale differences sequences or more general dependent sequences, and $V_n= {\rm Var}(S_n)$. Applications to line…
A univariate Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes process has wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with depe…
Let $X(t)=(X_1(t), \dots, X_n(t)), t\in \mathcal{T}\subset \mathbb{R} $ be a centered vector-valued Gaussian process with independent components and continuous trajectories, and $h(t)=(h_1(t),\dots, h_n(t)), t\in \mathcal{T} $ be a vector-valued continuous function. We investigate the asymptotics of $$\mathbb{P}\left(\sup_{t\in \mathcal{T} } \min_{1\leq i\leq n}(X_i(t)+h_i(t))>u\right)$$ as $u\to\infty$. As an illustration to the derived results we analyze two important classes of $X(t)$: with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymp…
We build a sequence of empirical measures on the space D(R_+,R^d) of R^d-valued càdlàg functions on R_+ in order to approximate the law of a stationary R^d-valued Markov and Feller process (X_t). We obtain some general results of convergence of this sequence. Then, we apply them to Brownian diffusions and solutions to Lévy driven SDE's under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure gives an efficient way of option pricing in stochastic volatility models.
We consider the Fleming--Viot particle system associated with a continuous-time Markov chain in a finite space. Assuming irreducibility, it is known that the particle system possesses a unique stationary distribution, under which its empirical measure converges to the quasistationary distribution of the Markov chain. We complement this Law of Large Numbers with a Central Limit Theorem. Our proof essentially relies on elementary computations on the infinitesimal generator of the Fleming--Viot particle system, and involves the so-called $π$-return process in the expression of the asymptotic vari…
In this paper, we focus on studying central limit theorems for functionals of some specific stationary random processes. In classical probability theory, it is well-known that for non-linear functionals of stationary Gaussian sequences, we can get a central-limit result via Hermite polynomials and the diagram formula for cumulants. In this paper, the main result is an analogous central limit theorem, in a free probability setting, for real-valued functionals of a stationary semicircular sequence with long-range dependence, namely the correlation function of the underlying time series tends to …
We study the asymptotic behavior of the weighted least squares estimators of the unknown parameters of bifurcating integer-valued autoregressive processes. Under suitable assumptions on the immigration, we establish the almost sure convergence of our estimators, together with the quadratic strong law and central limit theorems. All our investigation relies on asymptotic results for vector-valued martingales.
