An invariance principle for \(\phi\)-mixing sequences. (English) Zbl 0597.60018

Peligrad, Magda

Ann. Probab. 13, 1304-1313 (1985).

It is established that the sample paths \(S_ n=X_ 1+...+X_ n\), properly normalized, of a \(\phi\)-mixing sequence \((X_ n)\) converge to Brownian motion under the Lindeberg’s condition and under some stationarity assumptions. No mixing rate is required. This generalizes well-known results of I. A. Ibragimov and Yu. V. Linnik [Independent and stationary sequences of random variables. (1971; Zbl 0219.60027)] and I. A. Ibragimov [Teor. Veroyatn. Primen. 20, 134-140 (1975; Zbl 0335.60023)]. Unanswered remains the following conjecture: If \((X_ n)\) is strictly stationary with \(EX^ 2_ 1<\infty\) and \(\sigma^ 2_ n=ES^ 2_ n\to \infty\), does the invariance principle hold? The author shows that it is true, if and only if \(\lim \inf_{n\to \infty}\sigma^ 2_ n/n>0\) for every strictly stationary sequence \((X_ n)\).

Reviewer: G.Kersting

MSC:

60F05	Central limit and other weak theorems
60B10	Convergence of probability measures

Keywords:

phi-mixing sequence; Lindeberg’s condition; invariance principle; strictly stationary sequence

Citations:

Zbl 0219.60027; Zbl 0335.60023