Distributional limit theorems over a stationary Gaussian sequence of random vectors

Autoregressive Gaussian Random Vectors of First Order

Limit theorems for excursion sets of stationary random fields

We give an overview of the recent asymptotic results on the geometry of excursion sets of stationary random fields. Namely, we cover a number of limit theorems of central type for the volume of excursions of stationary (quasi–, positively or negatively) associated random fields with stochastically continuous realizations for a fixed excursion level. This class includes in particular Gaussian, P...

Strong limit theorems for partial sums of a random sequence

Let {ξj ; j ≥ 1} be a centered strictly stationary random sequence defined by S0 = 0, Sn = ∑n j=1 ξj and σ(n) = √ ES2 n, where σ(t), t > 0, is a nondecreasing continuous regularly varying function. Suppose that there exists n0 ≥ 1 such that, for any n ≥ n0 and 0 ≤ ε < 1, there exist positive constants c1 and c2 such that c1 e−(1+ε)x 2/2 ≤ P { |Sn| σ(n) ≥ x } ≤ c2 e −(1−ε)x2/2, x ≥ 1. Under some...

autoregressive gaussian random vectors of first order

Limit Theorems for Stationary Distributions

For every N > 1, let X,X, X, * * be a real-valued stationary process, and let AX = Xn Xn. Suppose that E(AXn XN)= O(FN) and var(AXN xN)= O(TN), where eN and 7N are positive null sequences. Limiting distributions of XN as N oc are obtained for the cases tN = e N and 7N = o(eN). These results are established by an extension of a method due to Moran. The theory is illustrated by a variety of appli...