多维局部平稳高斯过程最大值的联合渐近分布

$\{(X_1(t),\cdots,X_p(t)),0\leq t\leq T\}$为$p$维局部平稳高斯过程, 具有渐近中心化的均值$m_k(t)$和常数的方差, $M_k(T)=\sup\{X_k(t),0\leq t\leq T\},\;k=1,\cdots,p$, 当$T\rightarrow\infty$时, 本文在一定条件下获得了$M(T)=(M_1(T),\cdots,M_p(T))$的联合渐近分布.     

On almost sure max-limit theorems of complete and incomplete samples from stationary sequences

Let M _n denote the partial maximum of a strictly stationary sequence (X _n ). Suppose that some of the random variables of (X _n ) can be observed and let [(M)tilde]_ntilde M_n stand for the maximum of observed random variables from the set {X ₁, ..., X _n }. In this paper, the almost sure limit theorems related to random vector ([(M)tilde]_ntilde M_n , M _n ) are considered in terms of i.i.d. case. The related results are also extended to weakly dependent stationary Gaussian sequence as its covariance function satisfies some regular conditions.     

Condition for the Convergence of Maxima of Random Triangular Arrays

We consider the convergence of the maxima of a triangular array of random variables. Sufficient and necessary conditions are discussed, assuming that the underlying distributions are twice differentiable. Our results extend those known for the iid. case.     

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

ABSTRACT

In this paper, for centred homogeneous Gaussian random fields the joint limiting distributions of normalized maxima and minima over continuous time and uniform grids are investigated. It is shown that maxima and minima are asymptotic dependent for strongly dependent homogeneous Gaussian random field with the choice of sparse grid, Pickands' grid or dense grid, while for the weakly dependent Gaussian random field maxima and minima are asymptotically independent.     

On the sequence of partial maxima of some random sequences

Let {X_n, n ? 1} be a sequence of identically distributed random variables, Z_n = max {X₁,…, X_n} and {u_n, n ? 1 } an increasing sequence of real numbers. Under certain additional requirements, necessary and sufficient conditions are given to have, with probability one, an infinite number of crossings of {Z_n} with respect to {u_n}, in two cases: (1) The X_n's are independent, (2) {X_n} is stationary Gaussian and satisfies a mixing condition.     

On Piterbarg’s max-discretisation theorem for multivariate stationary Gaussian processes

Let {X(t),t≥0}$\{X\left(t\right),t\ge 0\}$ be a stationary Gaussian process with zero-mean and unit variance. A deep result derived in Piterbarg (2004)  [23], which we refer to as Piterbarg’s max-discretisation theorem gives the joint asymptotic behaviour (T→∞$T\to \infty$) of the continuous time maximum M(T)=_maxt∈[0,T]X(t)$M\left(T\right)={max}_{t\in [0,T]}X\left(t\right)$, and the maximum M^δ(T)=_maxt∈R(δ)X(t)$M^{\delta }\left(T\right)={max}_{t\in \mathfrak{R}\left(\delta \right)}X\left(t\right)$, with R(δ)⊂[0,T]$\mathfrak{R}\left(\delta \right)\subset [0,T]$ a uniform grid of points of distance δ=δ(T)$\delta =\delta \left(T\right)$. Under some asymptotic restrictions on the correlation function Piterbarg’s max-discretisation theorem shows that for the limit result it is important to know the speed δ(T)$\delta \left(T\right)$ approaches 0 as T→∞$T\to \infty$. The present contribution derives the aforementioned theorem for multivariate stationary Gaussian processes.     

Extreme value distribution for normalized sums from stationary Gaussian sequences

Let {X_i, i?0} be a sequence of independent identically distributed random variables with finite absolute third moment. Then Darling and Erdös have shown that

for -∞<t<∞ where and $\text{X}{}_{\mathrm{n}}\text{= (2 ln ln n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. The result is extended to dependent sequences but assuming that {X_i} is a standard stationary Gaussian sequence with covariance function {r_i}. When {X_i} is moderately dependent (e.g. when $\text{v(∑}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{X}{}_{\mathrm{i}}\text{) ? n}{}_{\mathrm{a}}\text{, 0 < a < 2)}$ we get

where H_a is a constant. In the strongly dependent case (e.g. when $\text{v(∑}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{X}{}_{\mathrm{i}}\text{) ? n}{}^2\text{r(n))}$ we get

for-∞<t<∞.     

Extremes in autoregressive processes with uniform marginal distributions

Chernick (1981) derives a limit theorem for the maximum term for a class of first order autoregressive processes with uniform marginal distributions. The parameter for these processes is equal to 1/r where r is an integer, r 2. Based on this limit theorem, the asymptotic distribution of the minimum term and the joint asymptotic distribution of the maximum and minimum terms in the sequence are obtained. Since the condition D′(u_n) of Leadbetter (1974) fails, the condition of Davis (1979), D′(v_n, u_n), also fails. Negatively correlated uniform sequences are shown to exist. Asymptotic distributions for the maximum and minimum terms in the sequence are derived and it is shown that the maximum and minimum are not asymptotically independent.     

Almost sure central limit theorem for strictly stationary processes

On any aperiodic measure preserving system, there exists a square integrable function such that the associated stationary process satifies the Almost Sure Central Limit Theorem.

     

Discrete wave-analysis of continuous stochastic processes

The behaviour of a continuous-time stochastic process in the neighbourhood of zero-crossings and local maxima is compared with the behaviour of a discrete sampled version of the same process.For regular processes, with finite crossing-rate or finite rate of local extremes, the behaviour of the sampled version approaches that of the continuous one as the sampling interval tends to zero. Especially the zero-crossing distance and the wave-length (i.e., the time from a local maximum to the next minimum) have asymptotically the same distributions in the discrete and the continuous case. Three numerical illustrations show that there is a good agreement even for rather big sampling intervals.For non-regular processes, with infinite crossing-rate, the sampling procedure can yield useful results. An example is given in which a small irregular disturbance is superposed over a regular process. The structure of the regular process is easily observable with a moderate sampling interval, but is completely hidden with a small interval.     

A mixture-type limit theorem for nonlinear functions of Gaussian sequences

We show that, for a certain class of nonlinear functions of Gaussian sequences, the limiting distribution of normalized sums of the nonlinear function values of a sequence is the convolution of a Gaussian distribution with another non-Gaussian distribution.     

Central Limit Theorems for the Number of Maxima and an Estimator of the Second Spectral Moment of a Stationary Gaussian Process, with Application to Hydroscience

Let X = (X_t, t 0) be a mean zero stationary Gaussian process with variance one, assumed to satisfy some conditions on its covariance function r. Central limit theorems and asymptotic variance formulas are provided for estimators of the square root of the second spectral moment of the process and for the number of maxima in an interval, with some applications in hydroscience. A consistent estimator of the asymptotic variance is proposed for the number of maxima.     

Conditions for the convergence in distribution of maxima of stationary normal processes

The asymptotic distribution of the maximum M_n=max_1?t?nξ_t in a stationary normal sequence ξ₁,ξ,… depends on the correlation r_t between ξ₀ and ξ_t. It is well known that if r_t log t → 0 as t → ∞ or if Σr²_t<∞, then the limiting distribution is the same as for a sequence of independent normal variables. Here it is shown that this also follows from a weaker condition, which only puts a restriction on the number of t-values for which rt log t islarge. The condition gives some insight into what is essential for this asymptotic behaviour of maxima. Similar results are obtained for a stationary normal process in continuous time.     

Almost sure limit theorems of extremes of complete and incomplete samples of stationary sequences

Let \((X_{n}^{\ast})\) be an independent identically distributed random sequence. Let \(M_{n}^{\ast}\) and \(m_{n}^{\ast}\) denote, respectively, the maximum and minimum of \(\{X_{1}^{\ast},\cdots,X_{n}^{\ast}\}\). Suppose that some of the random variables \(X_1^{\ast},X_2^{\ast},\cdots\) can be observed and let \(\widetilde{M}_n^{\ast}\) and \(\widetilde{m}_n^{\ast}\) denote, respectively, the maximum and minimum of the observed random variables from the set \(\{X_1^{\ast},\cdots,X_n^{\ast}\}\). In this paper, we consider the asymptotic joint limiting distribution and the almost sure limit theorems related to the random vector \((\widetilde{M}_n^{\ast}, \widetilde{m}_n^{\ast}, M_n^{\ast}, m_n^{\ast})\). The results are extended to weakly dependent stationary Gaussian sequences.     

Second-order expansion for the maximum of some stationary Gaussian sequences

We prove a second-order approximation formula for the distribution of the largest term among an infinite moving average Gaussian sequence. The second-order correction term depends on the autocovariance function only through the second largest autocovariance. Applications to Gaussian time series are discussed and a simulation study showed a substantial improvement over other approximations to the exact distribution of the maximum.     

Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences

Let be a sequence of d-dimensional stationary Gaussian vectors, and let denote the partial maxima of . Suppose that there are missing data in each component of and let denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector where the correlation and cross-correlation satisfy some dependence conditions.     

Almost sure convergence for the maxima of strongly dependent stationary Gaussian vector sequences

In this paper, we prove the almost sure limit theorem of the maxima for a kind of strongly dependent stationary Gaussian vector sequences.