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.     

The Second Appell Function for one Large Variable

We consider the Mellin convolution integral representation of the second Appell function given in [8]. Then, we apply the asymptotic method designed in [12] for this kind of integrals to derive new asymptotic expansions of the Appell function F ₂ for one large variable in terms of hypergeometric functions. For certain values of the parameters, some of these expansions involve logarithmic terms in the asymptotic variables. The accuracy of the approximations is illustrated with numerical experiments.     

Exact tail asymptotics in bivariate scale mixture models

Let (X, Y) = (RU ₁, RU ₂) be a given bivariate scale mixture random vector, with R > 0 independent of the bivariate random vector (U ₁, U ₂). In this paper we derive exact asymptotic expansions of the joint survivor probability of (X, Y) assuming that R has distribution function in the Gumbel max-domain of attraction, and (U ₁, U ₂) has a specific local asymptotic behaviour around some absorbing point. We apply our results to investigate the asymptotic behaviour of joint conditional excess distribution and the asymptotic independence for two models of bivariate scale mixture distributions.     

Asymptotic expansions in the central limit theorem for compound and Markov processes

Summary For independent identically distributed bivariate random vectors (X ₁, Y ₁), (X ₂, Y ₂), ... and for large t the distribution of X ₁ +...+ X _N(t) is approximated by asymptotic expansions. Here N(t) is the counting process with lifetimes Y ₁, Y ₂,.... Similar expansions are derived for multivariate X ₁. Furthermore, local asymptotic expansions are valid for the distribution of f(X ₁)+ ...+ f(X _N ) when N is large and nonrandom, and X _i , i=1, 2,..., is a discrete strongly mixing Markov chain.     

Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of ergodic distribution of the process X(t) are obtained when the random variable ζ₁, which is describing a discrete interference of chance, has a triangular distribution in the interval [s, S] with center (S + s)/2. Based on these results, the asymptotic expansions with three-term are obtained for the first four moments of the ergodic distribution of X(t), as a ≡ (S − s)/2 → ∞. Furthermore, the asymptotic expansions for the variance, skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter a.     

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n^?2, for the ₀F₀ function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root.     

Asymptotic Results for Tail Probabilities of Sums of Dependent and Heavy-Tailed Random Variables

Abstract Let X1,X2,...be a sequence of dependent and heavy-tailed random variables with distributions F1,F2,…. on (-∞,∞),and let т be a nonnegative integer-valued random variable independent of the seq...     

Uniform asymptotics of the eigenvalues and eigenfunctions of the Dirac system with an integrable potential

We consider the Dirac operator on a finite interval with a potential belonging to some set X completely bounded in the space L₁[0, π] and with strongly regular boundary conditions. We derive asymptotic formulas for the eigenvalues and eigenfunctions of the operator; moreover, the constants occurring in the estimates for the remainders depend on the boundary conditions and the set X alone.     

Asymptotic expansions in the integral and local limit theorems in banach spaces with applications to ω-statistics

LetB be a real separable Banach space and letX,X ₁,X ₂,...∈B denote a sequence of independent identically distributed random variables taking values inB. DenoteS _n =n ^?1/2(X ₁+...X _n ). Let π:B→R be a polynomial. We consider (truncated) Edgeworth expansions and other asymptotic expansions for the distribution function of the r.v. π(S _n ) with uniform and nonuniform bounds for the remainder terms. Expansions for the density of π(S _n ) and its higher order derivatives are derived as well. As an application of the general results we get expansions in the integral and local limit theorems for ω-statistics $$\omega _n^p (q)\mathop { = n^{{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} }\limits^\Delta \smallint _{(0,1)} \{ F_n (x) - x\} ^p q(x)dx$$ and investigate smoothness properties of their distribution functions. Herep≥2 is an even number,q: [0, 1]→[0, ∞] is a measurable weight function, andF _n denotes the empirical distribution function. Roughly speaking, we show that in order to get an asymptotic expansion with remainder termO(n ^?α), α<p/2, for the distribution function of the ω-statistic, it is sufficient thatq is nontrivial, i.e., mes{t∈(0, 1):q(t)≠0}>0. Expansions of arbitrary length are available provided the weight functionq is absolutely continuous and positive on an nonempty subinterval of (0, 1). Similar results hold for the density of the distribution function and its derivatives providedq satisfies certain very mild smoothness condition and is bounded away from zero. The last condition is essential since the distribution function of the ω-statistic has no density whenq is vanishing on an nonempty subinterval of (0, 1).     

Asymptotics for the partial sum and its maximum of dependent random variables<Superscript>*</Superscript>

Let X ₁,…,X _n be pairwise asymptotically independent or pairwise upper extended negatively dependent real-valued random variables. Under the condition that the distribution of the maximum of X ₁,…,X _n belongs to some subclass of heavy-tailed distributions, we investigate the asymptotic behavior of the partial sum and its maximum generated by dependent X ₁,…,X _n . As an application, we consider a discrete-time risk model with insurance and financial risks and derive the asymptotics for the finite-time ruin probability.     

Asymptotics of random contractions

In this paper we discuss the asymptotic behaviour of random contractions X=RS, where R, with distribution function F, is a positive random variable independent of S∈(0,1). Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.     

Saddlepoint expansions for sums of Markov dependent variables on a continuous state space

Summary Based on the conjugate kernel studied in Iscoe et al. (1985) we derive saddlepoint expansions for either the density or distribution function of a sumf(X ₁)+...+f(X _n ), where theX _i 's constitute a Markov chain. The chain is assumed to satisfy a strong recurrence condition which makes the results here very similar to the classical results for i.i.d. variables. In particular we establish also conditions under which the expansions hold uniformly over the range of the saddlepoint. Expansions are also derived for sums of the formf(X ₁,X ₀)+f(X ₂,X ₁)+...+f(X _n ,X _n–1) although the uniformity result just mentioned does not generalize.     

Laplace's method on a computer algebra system with an application to the real valued modified Bessel functions

We examine a Maple implementation of two distinct approaches to Laplace's method used to obtain asymptotic expansions of Laplace-type integrals. One algorithm uses power series reversion, whereas the other expands all quantities in Taylor or Puiseux series. These algorithms are used to derive asymptotic expansions for the real valued modified Bessel functions of pure imaginary order and real argument that mimic the well-known corresponding expansions for the unmodified Bessel functions.     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.     

Tail behavior of the product of two dependent random variables with applications to risk theory

Let the random vector (X,Y) follow a bivariate Sarmanov distribution, where X is real-valued and Y is nonnegative. In this paper we investigate the impact of such a dependence structure between X and Y on the tail behavior of their product Z?=?XY. When X has a regularly varying tail, we establish an asymptotic formula, which extends Breiman’s theorem. Based on the obtained result, we consider a discrete-time insurance risk model with dependent insurance and financial risks, and derive the asymptotic and uniformly asymptotic behavior for the (in)finite-time ruin probabilities.     

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.     

Random matrix products and measures on projective spaces

The asymptotic behavior of ‖X _n X _n ?1…X ₁υ‖ is studied for independent matrix-valued random variablesX _n . The main tool is the use of auxiliary measures in projective space and the study of markov processes on projective space.     

Second order regular variation and conditional tail expectation of multiple risks

For the purpose of risk management, the study of tail behavior of multiple risks is more relevant than the study of their overall distributions. Asymptotic study assuming that each marginal risk goes to infinity is more mathematically tractable and has also uncovered some interesting performance of risk measures and relationships between risk measures by their first order approximations. However, the first order approximation is only a crude way to understand tail behavior of multiple risks, and especially for sub-extremal risks. In this paper, we conduct asymptotic analysis on conditional tail expectation (CTE) under the condition of second order regular variation (2RV). First, the closed-form second order approximation of CTE is obtained for the univariate case. Then CTE of the form E[X₁∣g(X₁,…,X_d)>t], as t→∞, is studied, where g is a loss aggregating function and (X₁,…,X_d)?(RT₁,…,RT_d) with R independent of (T₁,…,T_d) and the survivor function of R satisfying the condition of 2RV. Closed-form second order approximations of CTE for this multivariate form have been derived in terms of corresponding value at risk. For both the univariate and multivariate cases, we find that the first order approximation is affected by only the regular variation index −α of marginal survivor functions, while the second order approximation is influenced by both the parameters for first and second order regular variation, and the rate of convergence to the first order approximation is dominated by the second order parameter only. We have also shown that the 2RV condition and the assumptions for the multivariate form are satisfied by many parametric distribution families, and thus the closed-form approximations would be useful for applications. Those closed-form results extend the study of Zhu and Li (submitted for publication).     

Asymptotic expansions for the zeros of certain special functions

We derive asymptotic expansions for the zeros of the cosine-integral Ci(x) and the Struve function $\text{H}{}_0\text{(x)}$, and extend the available formulae for the zeros of Kelvin functions. Numerical evidence is provided to illustrate the accuracy of the expansions.     

Asymptotic law of distribution of primes of special form

Let N₀ be the set of natural numbers whose binary expansions have an even number of 1’s, and let N₁ = N\N₀. In this paper, we obtain asymptotic formulas for the number of primes p not exceeding X and such that p ∈ N_i, p + 1 ∈ N_j, where i and j take values 0 and 1 independently of each other.