Given $\{X_k\}$ is a martingale difference sequence. And given another $\{Y_k\}$ which has dependency within the sequence. Assume $\{X_k\}$ is independent with $\{Y_k\}$, we study the properties of the sums of product of two sequences $\sum_{k=1}^{n} X_k Y_k$. We obtain product-CLT, a modification of classical central limit theorem, which can be useful in the study of random projections. We also obtain the rate of convergence which is similar to the Berry-Essen theorem in the classical CLT.
翻译:鉴于 $X_k ⁇ $是一个马丁加尔差异序列。 而且如果给另外的$Y_k ⁇ $在序列中具有依赖性, 假设$X_k ⁇ $与$Y_k ⁇ $是独立的, 我们研究两个序列的产品总和的属性 $sum@k=1 ⁇ n} X_k Y_k$。 我们获得了产品- CLT, 这是对经典中央界限理论的修改, 可用于随机预测的研究。 我们还获得了类似于经典CLT中的Berry- Essen理论的趋同率 。