### Almost sure convergence for self-normalized products of sums of partial sums of $\rho^-$-mixing sequences

#### Abstract

Let $X, X_1, X_2,\ldots$ be a stationary sequence of $\rho^-$-mixing positive random variables. A universal result in the area of almost

sure central limit theorems for the self-normalized products of sums

of partial sums $(\prod_{j=1}^{k}(T_j/(j(j+1)\mu/2)))^{\mu/(\beta V_k)}$ is established, where: $T_j=\sum_{i=1}^{j}S_i, S_i=\sum_{k=1}^{i}X_k, V_k=\sqrt{\sum_{i=1}^kX^2_i}, \mu=\mathbb{E}X, \beta>0$. Our results generalize and

improve those on almost sure central limit theorems obtained by previous authors from the independent case to $\rho^-$-mixing sequences and from partial sums case to self-normalized products of sums of partial sums.

sure central limit theorems for the self-normalized products of sums

of partial sums $(\prod_{j=1}^{k}(T_j/(j(j+1)\mu/2)))^{\mu/(\beta V_k)}$ is established, where: $T_j=\sum_{i=1}^{j}S_i, S_i=\sum_{k=1}^{i}X_k, V_k=\sqrt{\sum_{i=1}^kX^2_i}, \mu=\mathbb{E}X, \beta>0$. Our results generalize and

improve those on almost sure central limit theorems obtained by previous authors from the independent case to $\rho^-$-mixing sequences and from partial sums case to self-normalized products of sums of partial sums.

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