Almost sure central limit theorem for self-normalized products of the some partial sums of ρ−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho^{-}$\end{document}-mixing sequences

Let {X,Xn}n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{X, X_{n}\}_{n\in N}$\end{document} be a strictly stationary ρ−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho^{-}$\end{document}-mixing sequence of positive random variables, under the suitable conditions, we get the almost sure central limit theorem for the products of the some partial sums (∏i=1kSk,i(k−1)nμn)μβVk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$({\frac{\prod_{i=1}^{k}S_{k,i}}{(k-1)^{n}\mu ^{n}} )^{\frac{\mu}{\beta V_{k}}} }$\end{document}, where β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta>0$\end{document} is a constant, and E(X)=μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathrm{E}}(X)=\mu$\end{document}, Sk,i=∑j=1kXj−Xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i}$\end{document}, 1≤i≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\le i\le k$\end{document}, Vk2=∑i=1k(Xi−μ)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V_{k}^{2}=\sum_{i=1}^{k}(X_{i}-\mu)^{2}$\end{document}.


Introduction and main result
In 1988, Brosamler [1] and Schatte [2] proposed the almost sure central limit theorem (ASCLT) for the sequence of i.i.d. random variables. On the basis of i.i.d., Khurelbaatar and Grzegorz [3] got the ASCLT for the products of the some partial sums of random variables. In 2008, Miao [4] gave a new form of ASCLT for products of some partial sums.
Theorem A ( [4]) Let {X, X n } n∈N be a sequence of i.i.d. positive square integrable random variables with E(X 1 ) = μ, Var(X 1 ) = σ 2 > 0 and the coefficient of variation γ = σ μ . Denote the S k,i = k j=1 X j -X i , 1 ≤ i ≤ k. Then, for ∀x ∈ R, where F(·) is the distribution function of the random variables e N , N is a standard normal random variable.
The precise definition of ρ --mixing random variables was introduced initially by Zhang and Wang [5] in 1999. Obviously, ρ --mixing random variables include NA and ρ * -mixing random variables, which have a lot of applications, their limit properties have aroused wide interest recently, and a lot of results have been obtained by many authors. In 2005, Zhou [6] proved the almost central limit theorem of the ρ --mixing sequence. The almost sure central limit theorem for products of the partial sums of ρ --mixing sequences was given by Tan [7] in 2012. Because the denominator of the self-normalized partial sums contains random variables, this brings about difficulties to the study of the self-normalized form limit theorem of the ρ --mixing sequence. At present, there are very few results of this kind. In this paper, we extend Theorem A, and get the almost sure central limit theorem for self-normalized products of the some partial sums of ρ --mixing sequences.
Throughout this paper, a n ∼ b n means lim n→∞ a n b n = 1, and C denotes a positive constant, which may take different values whenever it appears in different expressions, and log x = ln(x ∨ e). We assume {X, X n } n∈N is a strictly stationary sequence of ρ --mixing random variables, and we denote Y i = X iμ. For apparently, δ 2 n = δ 2 n,1 + δ 2 n,2 , E(V 2 n ) = nδ 2 n = nδ 2 n,1 + nδ 2 n,2 . Our main theorem is as follows.
Theorem 1 Let {X, X n } n∈N be a strictly stationary ρ --mixing sequence of positive random variables with EX = μ > 0, and for some r > 2, we have 0 < E|X| r < ∞. Denote Suppose 0 ≤ α < 1 2 , and let then, for ∀x ∈ R, we have where F(·) is the distribution function of the random variables e N , N is a standard normal random variable.
Corollary 2 If {X n , n ≥ 1} is a sequence of strictly stationary independent positive random variables then one has (a 3 ) and β = 1.

Some lemmas
We will need the following lemmas.

Lemma 2.2 ([9])
Let {X, X n } n∈N be a sequence of ρ --mixing random variables, with then there is a positive constant C = C(q, ρ -(·)) only depending on q and ρ -(·) such that

Lemma 2.5 If the assumptions of Theorem 1 hold, then
where d k and D k is defined as (1) and f is real, bounded, absolutely continuous function on R.
Proof Firstly, we prove (4), by the property of ρ --mixing sequence, we know that {Ȳ ni } n≥1,i≤n is a ρ --mixing sequence; using Lemma 2.1 in [7], the condition (a 2 ), (a 3 ), and hence, for any g(x) which is a bounded function with bounded continuous derivative, we have by the Toeplitz lemma, we get On the other hand, from Theorem 7.1 of [11] and Sect. 2 of [12], we know that (4) is equivalent to hence, to prove (4), it suffices to prove noting that First we estimate I 1 ; we know that g is a bounded Lipschitz function, i.e., there exists a constant C such that for any x, y ∈ R, since {Ȳ ni } n≥1,i≤n also is a ρ --mixing sequence; we use the condition δ 2 l → E(Y 2 ) < ∞, l → ∞, and Lemma 2.2, to get Next we estimate I 2 ; by Lemma 2.2, we have Var T l,l -ET l,l -(T 2k,l -ET 2k,l ) By the definition of a ρ --mixing sequence, EY 2 < ∞, and Lemma 2.3, we have By X 2,1 ≤ r/(r -2) X r , r > 2 (see p. 254 of [10] or p. 251 of [13]), Minkowski inequality, Lemma 2.2, and the Hölder inequality, we get Hence Combining with (7)-(9), (3) holds, and by (a 4 ), Lemma 2.4, (6) holds, then (4) is true.