Weak invariance principle for mixing sequences in the domain of attraction of normal law. (English) Zbl 1240.60034

Balan, Raluca M.; Kulik, Rafał

Stud. Sci. Math. Hung. 46, No. 3, 329-343 (2009).

The weak invariance principle in a self-normalized form is proved for strictly stationary sequences of non-degenerate random variables \((X_n, n\geq 0)\) under the following set of assumptions: (1) the truncated variance function \(EX_1^2 1_{\{| X_1| \leq x\}}\) is slowly varying at infinity as a function of \(x\); (2) the \(\phi\)-mixing coefficient has a logarithmic decay and \(\phi(1) < 1/4\); (3) the variance of a specially truncated sum is equivalent to the sum of corresponding truncated variances with a positive multiplier, as a function of \(n\); the variance itself may diverge.
The main result is a new interesting development in the mainstream of the papers [R. C. Bradley, Ann. Probab. 16, No.  1, 313–332 (1988; Zbl 0643.60018); E. Giné, F. Götze and D. M. Mason, Ann. Probab. 25, No.  3, 1514–1531 (1997; Zbl 0958.60023); Q. Shao, Chin. Ann. Math., Ser. B 14, No.  1, 27–42 (1993; Zbl 0777.60028); M. Csörgő, B. Szyszkowicz and Q. Wang, Ann. Probab. 31, No.  3, 1228–1240 (2003; Zbl 1045.60020); R. Balan and I.-M. Zamfirescu, Electron. Commun. Probab. 11, 11–23, electronic only (2006; Zbl 1109.60023)].

Reviewer: Alexander Yu. Veretennikov (Leeds)

MSC:

60F05	Central limit and other weak theorems
60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60F17	Functional limit theorems; invariance principles

Keywords:

weak invariance principle; stationary \(\phi\)-mixing sequence; truncation; self-normalized central limit theorem

Citations:

Zbl 0643.60018; Zbl 0958.60023; Zbl 0777.60028; Zbl 1045.60020; Zbl 1109.60023