The invariance principle for nonstationary sequences of associated random variables. (English) Zbl 0707.60004

Matuła, Przemysław; Rychlik, Zdzisław

Ann. Inst. Henri Poincaré, Probab. Stat. 26, No. 3, 387-397 (1990).

A finite collection of random variables \(X_ 1,...,X_ n\) is said to be associated, if for any two coordinatewise non-decreasing functions \(f_ 1,f_ 2\) on \(R^ n\) such that the random variables \(f_ 1(X_ 1,...,X_ n)\), \(f_ 2(X_ 1,...,X_ n)\) are square integrable, their cross-covariance is non-negative. A sequence \((X_ n)\) of random variables is said to be associated if any finite collection is associated.
The invariance principle for associated sequences of random variables has been recently studied extensively. However, in all previous papers some sort of stationarity of \((X_ n)\) has been required. The authors of this paper succeeded in finding a general sufficient condition for the validity of the invariance principle for associated sequences. In their setting the approximating step-functions are built by means of arbitrary, infinitely increasing sequences of positive numbers \((k_ n):\) \[ W_ n(t)=S_{m_ n(t)}/S_ n,\quad t\in [0,1],\quad m_ n(t)=m(t\cdot k_ n),\quad m(\tau)=\max \{i:\;k_ i\leq \tau \},\quad \tau \geq 0. \]

Reviewer: S.A.Chobanjan

MSC:

60B10	Convergence of probability measures
60F17	Functional limit theorems; invariance principles
60F05	Central limit and other weak theorems
60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

Keywords:

associated sequence of random variables; invariance principle