A strong law of large numbers for generalized almost sure central limit theorems. (English) Zbl 1074.60036

Giuliano Antonini, Rita; Pratelli, Luca

Teor. Jmovirn. Mat. Stat. 70, 1-8 (2004) and Theory Probab. Math. Stat. 70, 1-9 (2005).

Let \(X_1\),…,\(X_n\) be a sequence of measurable integrable random variables. The main result of the paper is the following theorem: Let \(\varphi\) be an increasing positive function on \(R^{+}\) with \(\sup\varphi=\infty\) and let \(\psi\) be a decreasing function on \(R^{+}\) with \(\int_0^\infty\psi(u)du<\infty\). If for any \(i<j\) \[ \text{Cov}(X_i,X_j)| \leq \psi(\log(\varphi(i)-\varphi(j)+1)), \] then \[ U_n={1\over \varphi(n)}\sum_{k=1}^n(\varphi(k)-\varphi(k-1))(X_k- E X_k)\to 0\text{ a.s. and in }L_2. \] Moreover, if \(\int_0^\infty\psi(u)\exp(u)du<\infty\), then for \(\beta>3\), \(U_n[\varphi(n)/\log^\beta(\varphi(n))]^{1/2}\to 0\) a.s. and in \(L_2\).
This result is applied to demonstrate the almost sure central limit theorem for i.i.d. sequences.

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

60F15	Strong limit theorems
60F05	Central limit and other weak theorems

Keywords:

almost orthogonal random sequence; weak dependence