Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables
© Daxiang Ye and Qunying Wu. 2011
Received: 19 September 2010
Accepted: 26 January 2011
Published: 15 February 2011
We give here an almost sure central limit theorem for product of sums of strongly mixing positive random variables.
1. Introduction and Results
Khurelbaatar and Rempala  gave an ASCLT for product of partial sums of i.i.d. random variables as follows.
Recently, Jin  had proved that (1.1) holds under appropriate conditions for strongly mixing positive random variables and gave an ASCLT for product of partial sums of strongly mixing as follows.
The sequence in (1.3) is called weight. Under the conditions of Theorem 1.2, it is easy to see that (1.3) holds for every sequence with and . Clearly, the larger the weight sequence is, the stronger is the result (1.3).
We first give an ASCLT for strongly mixing positive random variables.
2. The Proofs
To prove theorems, we need the following lemmas.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
2.2. Proof of Theorem 1.4
2.3. Proof of Theorem 1.3
and thus (2.36) implies (2.37).
This work is supported by the National Natural Science Foundation of China (11061012), Innovation Project of Guangxi Graduate Education (200910596020M29).
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