Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) Σ_t=1ⁿ X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

A random measure model for the emission of pollutants by vehicles on a highway

Consider a collection of vehicles on a highway. The velocity process of the k^th vehicle is a Markov process with continuous paths independent of the velocity processes of the other vehicles but with the same probability law. The discharge of pollutants by the k^th vehicle is described by a nondecreasing right continuous process whose probability law depends on the path of the vehicle in such a way that the discharge process has independent increments given the velocity process of the vehicle. The limiting behavior of the total amount of pollutants discharged in an interval of the highway by time t by all the vehicles is studied as t → ∞.     

Density of Space-Time Distribution of Brownian First Hitting of a Disc and a Ball

We compute the joint distribution of the site and the time at which a d-dimensional standard Brownian motion ((B˙t)) hits the surface of the ball ((U(a) ={—x—<a})) for the first time. The asymptotic form of its density is obtained when either the hitting time or the starting site ((B˙0)) becomes large. Our results entail that if Brownian motion is started at ((x)) and conditioned to hit ((U(a))), at time t, the distribution of the hitting site approaches the uniform distribution or the point mass at ((ax/—x—)) according as ((—x—/t)) tends to zero or infinity; in each case we provide a precise asymptotic estimate of the density. In the case when ((—x—/t)) tends to a positive constant we show the convergence of the density and derive an analytic expression of the limit density.     

A strong law of large numbers for vector gaussian martingales and a statistical application in linear regression

For a d-dimensional gaussian martingae M with tensor increasingprocess we prove that <M>⁺_tM_t converges in R^d with probability 1 as t → ∞ and the limit is zero a.s. iff tr <M>⁺_t tends to zero. We apply this result to study the strong consistency of estimates in a linear regression model.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

Maximum and minimum of one-dimensional diffusions

Let M_t be the maximum of a recurrent one-dimensional diffusion up till time t. Under appropriate conditions, there exists a distribution function F such that |P(M_t?x) ? F^t(x)|→0as t and x go to infinity. This reduces the asymptotic behavior of the maximum to that of the maximum of independent and identically distributed random variables with distribution function F. A new proof of this fact is given which is based on a time change of the Ornstein-Uhlenbeck process. Using this technique, the asymptotic independence of the maximum and minimum is also established. Moreover, this method allows one to construct stationary processes in which the limiting behavior of M_t is essentially unaffected by the stationary distribution. That is, there may be no relationship between the distribution F above and the marginal distribution of the process.     

Stochastic processes with independent increments,taking values in a Hilbert space

Let $\text{H}$ be a real separable Hilbert space; let X(t), t?[0, 1], be a separable, stochastically continuous, $\text{H}$-valued stochastic process with independent increments. Then a decomposition of X(t) into a uniformly convergent sum of independent processes is found. In this decomposition one of the processes is Gaussian with continuous sample functions, and the remainder of the processes have sample functions whose discontinuities correspond to those of certain real-valued Poisson processes. The decomposition of X(t) leads to a Levy-Khintchine representation of the characteristic functional of X(t). In addition, the case when X(t) has finite variance is explored, and, as a consequence of the above decomposition, a Kolmogorov-type representation of the characteristic functional of X(t) is derived.     

Characterizations of the Poisson process as a renewal process via two conditional moments

Given two independent positive random variables, under some minor conditions, it is known that fromE(X^rX+Y)=a(X+Y)^r andE(X^sX+Y)=b(X+Y)^s, for certain pairs ofr ands, wherea andb are two constants, we can characterizeX andY to have gamma distributions. Inspired by this, in this article we will characterize the Poisson process among the class of renewal processes via two conditional moments. More precisely, let {A(t), t0} be a renewal process, with {S _k, k1} the sequence of arrival times, andF the common distribution function of the inter-arrival times. We prove that for some fixedn andk, kn, ifE(S _k ^r A(t)=n)=at^r andE(S _k ^s A(t)=n)=bt^s, for certain pairs ofr ands, wherea andb are independent oft, then {A(t), t0} has to be a Poisson process. We also give some corresponding results about characterizingFto be geometric whenF is discrete.Support for this research was provided in part by the National Science Council of the Republic of China, Grant No. NSC 81-0208-M110-06.     

Optimal search strategies for Wienér processes

Let n independent Wiener processes be given. We assume that the following information is known about these processes; one process has drift μt, the remaining n - 1 processes have drift zero, all n processes have common variance σ²t, and we assume that a prior probability distribution over the n processes is given to identify the process with drift μt. A searcher is permitted to observe the increments of one process at a time with the object of identifying (with probability 1 - λ of correct selection) the process with drift μt.The authors define a natural class of search strategies and show that the strategy within this class which minimizes the total search time is the strategy which, whenever possible, searches the process which currently has the largest posterior probability of being the one with drift μt.     

On the Rate of Multivariate Poisson Convergence

The distribution of the sum of independent nonidentically distributed Bernoulli random vectors inR^kis approximated by a multivariate Poisson distribution. By using a multivariate adaption of Kerstan's (1964,Z. Wahrsch. verw. Gebiete2, 173–179) method, we prove a conjecture of Barbour (1988,J. Appl. Probab.25A, 175–184) on removing a log-term in the upper bound of the total variation distance. Second-order approximations are included.     

Recovering a Family of Two-Dimensional Gaussian Variables from the Minimum Process

Suppose that {(X _t, Y _t): t>}0 is a family of two independent Gaussian random variables with means m ₁(t) and m ₂(t) and variances σ ² ₁(t) and σ ² ₂(t). If at every time t>0 the first and second moment of the minimum process X _t∧Y _t are known, are the parameters governing these four moment functions uniquely determined ? We answer this question in the negative for a large class of Gaussian families including the “Brownian” case. Except for some degenerate situation where one variance function dominates the other, in which case the recovery of the parameters is fully successful, the second moment of the minimum process does not provide any additional clues on identifying the parameters.     

Processes of <Emphasis Type="Italic">r</Emphasis><Superscript><Emphasis Type="Italic">t</Emphasis><Emphasis Type="Italic">h</Emphasis></Superscript> largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an \(\mathbb {R}^{\infty }\)-valued stochastic process in r which we denote as M^(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M^(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M^(r) and M^(r) itself, after norming and centering. In continuous time, an analogous process Y^(r) based on a two-dimensional Poisson process on \(\mathbb {R}_{+}\times \mathbb {R}\) is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t >?0. This necessitates a different approach to the asymptotics in this case.     

On limit distributions of first crossing points of Gaussian sequences

Let {X_k, k?Z} be a stationary Gaussian sequence with EX₁ – 0, EX²_k = 1 and EX₀X_k = r_k. Define τ_x = inf{k: X_k >– βk} the first crossing point of the Gaussian sequence with the function – βt (β > 0). We consider limit distributions of τ_x as β→0, depending on the correlation function r_k. We generalize the results for crossing points τ_x = inf{k: X_k >β?(k)} with ?(– t)?t^γL(t) for t→∞, where γ > 0 and L(t) varies slowly.     

Small work-load mode in a queueing system with random nonstationary intensity

We consider a queueing system of M _t R|GI|1|∞ type with doubly stochastic Poisson arrival stream. The case of a small work load in such a system is studied. We derive an asymptotic expansion in the work-load smallness parameter of the distribution function of the virtual waiting time.     

Bootstrapping Autoregressive and Moving Average Parameter Estimates of Infinite Order Vector Autoregressive Processes

We consider anr-dimensional multivariate time series {y_t, tZ} which is generated by an infinite order vector autoregressive process. We show that a bootstrap procedure which works by generating time series replicates via an estimated finitek-order vector autoregressive process (k→∞ at an appropriate rate with the sample size) gives asymptotically valid approximations to the joint distribution of the growing set of estimated autoregressive coefficients and to the corresponding set of estimated moving average coefficients (impuls responses).     

On the classification of conditionally integrable evolution systems in (1 + 1) dimensions

We generalize earlier results of Fokas and Liu and find all locally analytic (1 + 1)-dimensional evolution equations of order n that admit an N-shock-type solution with N ≤ n + 1. For this, we develop a refinement of the technique from our earlier work, where we completely characterized all (1+1)-dimensional evolution systems u _t = F (x, t, u, ?u/?x,..., ?ⁿ u/? x ⁿ) that are conditionally invariant under a given generalized (Lie-Bäcklund) vector field Q(x, t, u, ?u/?x,..., ?^k u/?x ^k)?/?u under the assumption that the system of ODEs Q = 0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.     

Statistical inference of spectral estimation for continuous-time MA processes with finite second moments

In this paper we investigate a continuous-time MA (moving average) process (X _t ) _t≥0 sampled at an equally spaced time grid {Δ,2Δ, …, nΔ}, where the grid distance Δ > 0 is fixed and n denotes the number of observations, in the frequency domain. We derive for the process (X _kΔ) _k∈? with finite second moments the asymptotic behavior of the periodogram and of the lag-window spectral density estimator. The periodogram is not a consistent estimator for the spectral density of (X _kΔ) _k∈?. Different periodogram frequencies are asymptotically independent and exponentially distributed like for ARMA processes in discrete time. This result is basic for frequency bootstraps. In contrast, the lag-window spectral density estimator is a consistent estimator for the spectral density of (X _kΔ) _k∈? and moreover, it is asymptotically normally distributed.     

A class of viscous p-Laplace equation with nonlinear sources

In this paper, we prove the global existence of solutions to the initial boundary value problem of a viscous p-Laplace equation with nonlinear sources. The asymptotic behavior of solutions as the viscous coefficient k tends to zero is also investigated. In particular, we discuss the H¹-Galerkin finite element method for our problem and establish the error estimates for two semi-discrete approximate schemes.     

A phase transition behavior for Brownian motions interacting through their ranks

Consider a time-varying collection of n points on the positive real axis, modeled as Exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain ‘continuity at the edge’ condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a non-trivial Poisson–Dirichlet distribution. The underlying idea of the proof is inspired by Talagrand’s analysis of the low temperature phase of Derrida’s Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson–Dirichlet law.