Bulletin of the
Korean Mathematical Society BKMS

Let $\{X_j,\, j \ge 1\}$ be a strictly stationary $\phi$-mixing sequence of non-degenerate random variables with ${\bf E} X_1 =0$. In this paper, we establish a self-normalized weak invariance principle and a central limit theorem for the sequence $\{X_j\}$ under the condition that $L(x):={\bf E} X_1^2I\{|X_1|\le x\}\ \text{is a slowly varying function at $\infty$}$, without any higher moment conditions.

Keywords: self-normalized random variables, invariance principle, central limit theorem, mixing sequence

MSC numbers: Primary 60F15, 60F17