Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables
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Research Article Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables Daxiang Ye and Qunying Wu College of Science, Guilin University of Technology, Guilin 541004, China Correspondence should be addressed to Daxiang Ye, [email protected] Received 19 September 2010; Revised 1 January 2011; Accepted 26 January 2011 Academic Editor: Ondˇrej Doˇsly´ Copyright q 2011 D. Ye and Q. Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give here an almost sure central limit theorem for product of sums of strongly mixing positive random variables.
1. Introduction and Results In recent decades, there has been a lot of work on the almost sure central limit theorem ASCLT, we can refer to Brosamler 1, Schatte 2, Lacey and Philipp 3, and Peligrad and Shao 4. Khurelbaatar and Rempala 5 gave an ASCLT for product of partial sums of i.i.d. random variables as follows. Theorem 1.1. Let {Xn , n ≥ 1} be a sequence of i.i.d. positive random variables with EX1 μ > 0 and VarX1 σ 2 . Denote γ σ/μ the coefficient of variation. Then for any real x ⎛ ⎞ 1/γ √k k n S 1 1 ⎝ i i1 I lim ≤ x⎠ Fx n → ∞ ln n k k!μk k1
a.s.,
1.1
where Sn nk1 Xk , I∗ is the indicator function, F· is the distribution function of the random variable eN , and N is a standard normal variable. Recently, Jin 6 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.
2
Journal of Inequalities and Applications
Theorem 1.2. Let {Xn , n ≥ 1} be a sequence of identically distributed positive strongly mixing
random variable with EX1 μ > 0 and VarX1 σ 2 , dk 1/k, Dn nk1 dk . Denote by
γ σ/μ the coefficient of variation, σn2 Var nk1 Sk − kμ/kσ and Bn2 VarSn . Assume E|X1 |2δ < ∞ for some δ > 0,
−r
αn O n
Bn2 σ02 > 0, n→∞ n lim
2 for some r > 1 , δ
σ2 inf n > 0. n∈N n
1.2
Then for any real x ⎛ ⎞ 1/γσk k n S 1 i1 i lim dk I ⎝ ≤ x⎠ Fx a.s. k n → ∞ Dn k!μ k1
1.3
The sequence {dk , k ≥ 1} 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 dk∗ with 0 ≤ dk∗ ≤ dk and Dn∗ k≤n dk∗ → ∞ 7. Clearly, the larger the weight sequence dk is, the stronger is the result 1.3.
α In the following sections, let dk eln k /k, 0 ≤ α < 1/2, Dn nk1 dk , “ ” denote the inequality “≤” up to some universal constant. We first give an ASCLT for strongly mixing positive random variables. Theorem 1.3. Let {Xn , n ≥ 1} be a sequence of identically distributed positive strongly mixing random variable with EX1 μ > 0 and VarX1 σ 2 , dk and Dn as mentioned above. Denote
by γ σ/μ the coefficient of variation, σn2 Var nk1 Sk − kμ/kσ and Bn2 VarSn . Assume that 1.4
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