1 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])
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Let
\(\{X, X_{n}\}_{n\in N}\)
be a sequence of i.i.d. positive square integrable random variables with
\({\mathrm{E}}(X_{1})=\mu\), \(\operatorname{Var}(X_{1})=\sigma^{2}>0\)
and the coefficient of variation
\(\gamma=\frac{\sigma}{\mu}\). Denote the
\(S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i}\), \(1\leq i\leq k\). Then, for
\(\forall x \in R\),
where
\(F(\cdot)\)
is the distribution function of the random variables
\(e^{\mathscr {N}}\), \({\mathscr {N}}\)
is a standard normal random variable.
$$\lim_{N\to\infty}\frac{1}{\log N}\sum_{n=1}^{N} \frac{1}{n}\mathrm{I} \biggl[ \biggl(\frac{\prod_{k=1}^{n}S_{n,k}}{(n-1)^{n}\mu^{n}} \biggr)^{\frac {1}{\gamma\sqrt{n} }} \le{x} \biggr]=F(x)\quad \textit{a.s.}, $$
For random variables X, Y, define
where the sup is taken over all \(f,g\in \mathscr {C}\) such that \(\mathrm{E}(f(X))^{2}<\infty\) and \(\mathrm{E}(g(Y))^{2}<\infty\), and \(\mathscr {C}\) is a class of functions which are coordinatewise increasing.
$$\rho^{-}(X,Y)=0\vee\sup\frac{\operatorname{Cov}(f(X), g(Y))}{ (\operatorname{Var}f(X))^{\frac{1}{2}}(\operatorname{Var}g(Y))^{\frac{1}{2}}}, $$
Definition
([5])
A sequence \(\{X, X_{n}\}_{n\in N}\) is called \(\rho^{-}\)-mixing, if
where
\(\mathscr {C}\) is a class of functions which are coordinatewise increasing.
$$\rho^{-}(s)=\sup\bigl\{ \rho^{-}(S,T); S,T\subset{N}, \operatorname{dist}(S,T)\geq{s}\bigr\} \rightarrow{0}, \quad s\rightarrow \infty, $$
$$\rho^{-}(S,T)=0\vee{\sup}\biggl\{ \frac{\operatorname{Cov}\{f(X_{i},i\in {S}),g(X_{j},j\in{T})\}}{ \sqrt{\operatorname{Var}\{f(X_{i},i\in{S})\}\operatorname{Var}\{g(X_{j},j\in{T})\}} }, f,g\in{ \mathscr {C}}\biggr\} , $$
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The precise definition of \(\rho^{-}\)-mixing random variables was introduced initially by Zhang and Wang [5] in 1999. Obviously, \(\rho^{-}\)-mixing random variables include NA and \(\rho ^{*}\)-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 \(\rho^{-}\)-mixing sequence. The almost sure central limit theorem for products of the partial sums of \(\rho^{-}\)-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 \(\rho ^{-}\)-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 \(\rho^{-}\)-mixing sequences.
Throughout this paper, \(a_{n}\sim b_{n} \) means \(\lim_{n\to\infty }\frac{a_{n}}{b_{n} }=1\), and C denotes a positive constant, which may take different values whenever it appears in different expressions, and \(\log x=\ln(x\vee e)\). We assume \(\{X, X_{n}\}_{n\in N}\) is a strictly stationary sequence of \(\rho^{-}\)-mixing random variables, and we denote \(Y_{i}=X_{i}-\mu\).
For every \(1\leq i\leq k\leq n\), define
apparently, \(\delta_{n}^{2}=\delta_{n,1}^{2}+\delta_{n,2}^{2}\), \(\mathrm{E}(\bar{V}_{n}^{2})=n\delta_{n}^{2}=n\delta_{n,1}^{2}+n\delta _{n,2}^{2}\).
$$\begin{aligned}& \bar{Y}_{ni}= -\sqrt{n}\mathrm{I}(Y_{i}< - \sqrt{n})+Y_{i}\mathrm{I}\bigl( \vert Y_{i} \vert \le \sqrt{n} \bigr)+\sqrt{n}\mathrm{I}(Y_{i}>\sqrt{n}), \\& {T}_{k,n}=\sum_{i=1}^{k} \bar{Y}_{ni},\qquad V_{n}^{2}=\sum _{i=1}^{n}Y_{i}^{2},\qquad \bar{V}_{n}^{2}=\sum_{i=1}^{n} \bar{Y}_{ni}^{2}, \\& \bar{V}_{n,1}^{2}= \sum_{i=1}^{n}\bar{Y}_{ni}^{2} \mathrm{I}(Y_{i}\geq0),\qquad \bar{V}_{n,2}^{2}=\sum _{i=1}^{n}\bar{Y}_{ni}^{2} \mathrm{I}(Y_{i}< 0), \\& \sigma_{n}^{2}=\operatorname{Var}({T}_{n,n}), \qquad \delta_{n}^{2}=\mathrm{E}\bigl(\bar {Y}_{n1}^{2}\bigr),\qquad \delta_{n,1}^{2}= \mathrm{E}\bar{Y}_{n1}^{2}\mathrm{I}(Y_{1}\geq0), \qquad \delta_{n,2}^{2}=\mathrm{E}\bar{Y}_{n1}^{2} \mathrm{I}(Y_{1}< 0), \end{aligned}$$
Our main theorem is as follows.
Theorem 1
Let
\(\{X, X_{n}\}_{n\in N}\)
be a strictly stationary
\(\rho^{-}\)-mixing sequence of positive random variables with
\(\mathrm{E}X=\mu>0\), and for some
\(r>2\), we have
\(0<\mathrm{E}|X|^{r}<\infty\). Denote
\(S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i}\), \(1\leq i\leq k\)
and
\(Y=X-\mu\). Suppose that
Suppose
\(0\leq\alpha<\frac{1}{2}\), and let
then, for
\(\forall x \in R\), we have
where
\(F(\cdot)\)
is the distribution function of the random variables
\(e^{\mathscr {N}}\), \(\mathscr {N}\)
is a standard normal random variable.
(a1)
\(\mathrm{E}v(Y^{2}\mathrm{I}(Y\geq0))>0\), \(\mathrm{E}(Y^{2}\mathrm{I}(Y<0))>0\),
(a2)
\(\sigma_{1}^{2}=\mathrm{E}X_{1}^{2}+2\sum_{k=2}^{\infty}\operatorname{Cov}(X_{1},X_{k})>0\), \(\sum_{k=2}^{\infty}|\operatorname{Cov}(X_{1},X_{k})|<\infty\),
(a3)
\(\sigma_{k}^{2}\sim\beta^{2}k\delta_{k}^{2}\), for some
\(\beta>0\),
(a4)
\(\rho^{-}(n)=O(\log^{-\delta}n)\), \(\exists\delta>1\).
$$ d_{k}=\frac{\exp(\log^{\alpha}k)}{k},\qquad D_{n}=\sum _{k=1}^{n}d_{k}, $$
(1)
$$ \lim_{n\to\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{I} \biggl[ \biggl(\frac{\prod_{i=1}^{k}S_{k,i}}{(k-1)^{k}\mu^{k}} \biggr)^{\frac{\mu }{\beta V_{k} }}\le{x} \biggr]=F(x)\quad \textit{a.s.}, $$
(2)
Corollary 1
Corollary 2
If
\(\{X_{n}, n\ge1\}\)
is a sequence of strictly stationary independent positive random variables then one has (a3) and
\(\beta=1\).
2 Some lemmas
We will need the following lemmas.
Lemma 2.1
([7])
Let
\(\{X, X_{n}\}_{n\in N}\)
be a strictly stationary sequence of
\(\rho^{-}\)-mixing random variables with
\(\mathrm{E}X_{1}=0\), \(0<\mathrm{E}X_{1}^{2}<\infty\), \(\sigma_{1}^{2}=\mathrm{E}X_{1}^{2}+2\sum_{k=2}^{\infty}\operatorname{Cov}(X_{1},X_{k})>0\)
and
\(\sum_{k=2}^{\infty }|\operatorname{Cov}(X_{1},X_{k})|<\infty\), then, for
\(0< p<2\), we have
$$\frac{S_{n}}{n^{\frac{1}{p}}}\rightarrow0 ,\quad \textit{a.s.}, n\rightarrow\infty. $$
Lemma 2.2
([9])
Let
\(\{X, X_{n}\}_{n\in N}\)
be a sequence of
\(\rho^{-}\)-mixing random variables, with
then there is a positive constant
\(C=C(q, \rho^{-}(\cdot))\)
only depending on
q
and
\(\rho^{-}(\cdot)\)
such that
$$\mathrm{E}X_{n}=0,\qquad \mathrm{E}|X_{n}|^{q}< \infty,\quad \forall{n}\geq1, q\geq2, $$
$$\mathrm{E}\Bigl(\max_{1\leq{j}\leq{n}}|S_{j}|^{q} \Bigr)\leq{C}\Biggl\{ \sum_{i=1}^{n} \mathrm{E}|X_{i}|^{q}+ \Biggl(\sum _{i=1}^{n}\mathrm{E}X_{i}^{2} \Biggr)^{\frac{q}{2}}\Biggr\} . $$
Lemma 2.3
([10])
Suppose that
\(f_{1}(x)\)
and
\(f_{2}(y)\)
are real, bounded, absolutely continuous functions on
R
with
\(|f'_{1}(x)|\leq C_{1}\)
and
\(|f'_{2}(y)|\leq C_{2}\), then, for any random variables
X
and
Y,
where
\(\|X\|_{2,1}=\int_{0}^{\infty} (P(|X|>x) )^{\frac {1}{2}}\,dx\).
$$\bigl\vert \operatorname{Cov}\bigl(f_{1}(X), f_{2}(Y) \bigr) \bigr\vert \leq C_{1}C_{2}\bigl\{ - \operatorname{Cov}(X,Y)+8\rho ^{-}(X,Y)\|X\|_{2,1}\|Y \|_{2,1}\bigr\} , $$
Lemma 2.4
Let
\(\{\xi, \xi_{n}\}_{n\in N}\)
be a sequence of uniformly bounded random variables. If
\(\exists\delta>1\), \(\rho ^{-}(n)=O(\log^{-\delta}n)\), there exist constants
\(C>0\)
and
\(\varepsilon>0\), such that
then
$$ \vert \mathrm{E}\xi_{k}\xi_{l} \vert \leq C\biggl( \rho^{-}(k)+\biggl(\frac{k}{l}\biggr)^{\varepsilon }\biggr), \quad 1\leq2k< l, $$
(3)
$$\lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \xi_{k}=0, \quad \textit{a.s.} $$
Proof
See the proof of Theorem 1 in [7]. □
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.
$$\begin{aligned}& \lim_{n\to\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{I} \biggl[\frac {{T}_{k,k}-\mathrm{E}({T}_{k,k})}{\beta\delta_{k}\sqrt{k}}\le{x} \biggr]=\Phi(x) \quad \textit{a.s.}, \forall x \in R, \end{aligned}$$
(4)
$$\begin{aligned}& \lim_{n\to\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \biggl[f \biggl(\frac {\bar{V}_{k,l}^{2}}{k\delta_{k,l}^{2}} \biggr)-\mathrm{E}f \biggl(\frac{\bar {V}_{k,l}^{2}}{k\delta_{k,l}^{2}} \biggr) \biggr]=0 \quad \textit{a.s.}, l=1,2, \end{aligned}$$
(5)
Proof
Firstly, we prove (4), by the property of \(\rho ^{-}\)-mixing sequence, we know that \(\{\bar{Y}_{ni}\}_{n\geq1,i\leq n}\) is a \(\rho^{-}\)-mixing sequence; using Lemma 2.1 in [7], the condition (a2), (a3), and \(\beta>0\), \(\delta_{k}^{2}\rightarrow\mathrm{E}Y^{2}>0\), it follows that
hence, for any \(g(x)\) which is a bounded function with bounded continuous derivative, we have
by the Toeplitz lemma, we get
$$\frac{{T}_{k,k}-\mathrm{E}({T}_{k,k})}{\beta\delta_{k}\sqrt{{k}}} \stackrel{\mathrm{d}}{\rightarrow} \mathscr {N},\quad k \rightarrow\infty, $$
$$\mathrm{E}g \biggl(\frac{{T}_{k,k}-E({T}_{k,k})}{\beta\delta_{k}\sqrt {{k}}} \biggr)\rightarrow\mathrm{E}g(\mathscr {N}), \quad k\rightarrow\infty, $$
$$\lim_{n\to\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{E} \biggl[g \biggl(\frac{{T}_{k,k}-\mathrm{E}({T}_{k,k})}{\beta\delta_{k}\sqrt{k}} \biggr) \biggr]=\mathrm{E}\bigl(g( \mathscr {N})\bigr). $$
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
for every \(1\leq2k< l\), we have
$$\lim_{n\to\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k}g \biggl(\frac {{T}_{k,k}-\mathrm{E}({T}_{k,k})}{\beta\delta_{k}\sqrt{k}} \biggr)=\mathrm{E}\bigl(g(\mathscr {N})\bigr)\quad \mbox{a.s.}, $$
$$ \lim_{n\to\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \biggl[g \biggl(\frac {{T}_{k,k}-\mathrm{E}({T}_{k,k})}{\beta\delta_{k}\sqrt{k}} \biggr)-\mathrm{E} \biggl(g\frac{{T}_{k,k}-\mathrm{E}({T}_{k,k})}{\beta\delta_{k}\sqrt {k}} \biggr) \biggr]=0 \quad \mbox{a.s.}, $$
(6)
$$\xi_{k}=g \biggl(\frac{{T}_{k,k}-\mathrm{E}({T}_{k,k})}{\beta\delta_{k}\sqrt {k}} \biggr)-\mathrm{E} \biggl(g \biggl(\frac{{T}_{k,k}-\mathrm{E}({T}_{k,k})}{\beta\delta_{k}\sqrt{k}} \biggr) \biggr), $$
$$\begin{aligned} \vert \mathrm{E}\xi_{k}\xi_{l} \vert =& \biggl\vert \operatorname{Cov}\biggl(g\biggl(\frac {{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta\delta_{k}\sqrt{k}}\biggr),g\biggl( \frac {{T}_{l,l}-\mathrm{E}{T}_{l,l}}{\beta \delta_{l}\sqrt{l}}\biggr)\biggr) \biggr\vert \\ \leq& \biggl\vert \operatorname{Cov}\biggl(g\biggl(\frac{{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta \delta_{k}\sqrt{k}} \biggr),g\biggl(\frac{{T}_{l,l}-\mathrm{E}{T}_{l,l}}{\beta \delta_{l}\sqrt{l}}\biggr)-g\biggl(\frac{{T}_{l,l}-\mathrm{E}{T}_{l,l}-({T}_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta \delta_{l}\sqrt{l}}\biggr)\biggr) \biggr\vert \\ &{}+ \biggl\vert \operatorname{Cov}\biggl(g\biggl(\frac{{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta \delta_{k}\sqrt{k}} \biggr), g\biggl(\frac{{T}_{l,l}-\mathrm{E}{T}_{l,l}-({T}_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta\delta_{l}\sqrt{l}}\biggr)\biggr) \biggr\vert \\ =&I_{1}+I_{2}. \end{aligned}$$
(7)
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 \(x, y\in R\), since \(\{\bar{Y}_{ni}\}_{n\geq1,i\leq n}\) also is a \(\rho^{-}\)-mixing sequence; we use the condition \(\delta_{l}^{2}\rightarrow\mathrm{E}(Y^{2})<\infty \), \(l\rightarrow\infty\), and Lemma 2.2, to get
$$\bigl\vert g(x)-g(y) \bigr\vert \leq C|x-y| $$
$$\begin{aligned} I_{1} \leq& C\frac{\mathrm{E}|{T}_{2k,l}-\mathrm{E}{T}_{2k,l}|}{\sqrt{l}}\leq C \frac{\sqrt{\mathrm{E}({T}_{2k,l}-\mathrm{E}{T}_{2k,l})^{2}}}{\sqrt {l}} \\ \leq& \frac{C}{\sqrt{l}}\sqrt{ \sum_{i=1}^{2k} \mathrm{E}\bar{Y}_{l,i}^{2}}\leq \frac{C}{\sqrt{l}}\sqrt { \sum_{i=1}^{2k} \mathrm{E}Y^{2}}\leq C\biggl(\frac{k}{l}\biggr)^{\frac{1}{2}}. \end{aligned}$$
(8)
Next we estimate \(I_{2}\); by Lemma 2.2, we have
and
By the definition of a \(\rho^{-}\)-mixing sequence, \(\mathrm{E}Y^{2}<\infty \), and Lemma 2.3, we have
By \(\|X\|_{2,1}\leq r/(r-2)\|X\|_{r}\), \(r>2\) (see p. 254 of [10] or p. 251 of [13]), Minkowski inequality, Lemma 2.2, and the Hölder inequality, we get
similarly
Hence
Combining with (7)–(9), (3) holds, and by (a4), Lemma 2.4, (6) holds, then (4) is true.
$$\begin{aligned} \operatorname{Var} \biggl(\frac{{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta\delta_{k}\sqrt {k}} \biggr)&\leq\frac{C}{k} \operatorname{Var}({T}_{k,k}-\mathrm{E} {T}_{k,k}) \\ &\leq\frac{C}{k}\sum_{i=1}^{k} \mathrm{E}(\bar{Y}_{ki}-\mathrm{E}\bar {Y}_{ki})^{2} \leq\frac{C}{k}\sum_{i=1}^{k} \mathrm{E}(\bar {Y}_{ki})^{2}\leq\frac{C}{k} \cdot k \leq C \end{aligned}$$
$$\begin{aligned} \begin{aligned} \operatorname{Var} \biggl(\frac{{T}_{l,l}-\mathrm{E}{T}_{l,l}-({T}_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta \delta_{l}\sqrt{l}} \biggr)&\leq\frac{C}{l} \operatorname{Var}\bigl({T}_{l,l}-\mathrm{E} {T}_{l,l}-({T}_{2k,l}- \mathrm{E} {T}_{2k,l})\bigr) \\ &\leq\frac{C}{l}\sum_{i=2k+1}^{l} \mathrm{E}(\bar{Y}_{li}-\mathrm{E}\bar {Y}_{li})^{2}\leq\frac{C}{l}\Biggl(\sum_{i=1}^{l} \mathrm{E}\bar {Y}_{li}^{2}\Biggr) \\ &\leq\frac{C}{l} \cdot l \leq C . \end{aligned} \end{aligned}$$
$$\begin{aligned} I_{2} \leq&\biggl( -\operatorname{Cov} \biggl(\frac{{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta \delta_{k}\sqrt{k}}, \frac{{T}_{l,l}-\mathrm{E}{T}_{l,l}-({T}_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta \delta_{l}\sqrt{l}}\biggr) \\ &{}+8\rho^{-}\biggl(\frac{{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta\delta_{k}\sqrt {k}} ,\frac{{T}_{l,l}-\mathrm{E}{T}_{l,l}-({T}_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta \delta_{l}\sqrt{l}}\biggr) \\ &{}\cdot \biggl\Vert \frac{{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta\delta_{k}\sqrt{k}} \biggr\Vert _{2,1}\cdot \biggl\Vert \frac {{T}_{l,l}-\mathrm{E}{T}_{l,l}-({T}_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta \delta_{l}\sqrt{l}} \biggr\Vert _{2,1}\biggr) \\ \leq& C\rho^{-}(k) \biggl(\operatorname{Var}\biggl(\frac{{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta \delta_{k}\sqrt{k}} \biggr)\biggr)^{\frac{1}{2}}\cdot\biggl(\operatorname{Var} \biggl( \frac {{T}_{l,l}-\mathrm{E}{T}_{l,l}-({T}_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta \delta_{l}\sqrt{l}}\biggr)\biggr)^{\frac{1}{2}} \\ &{}+8\rho^{-}(k)\cdot \biggl\Vert \frac{{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta \delta_{k}\sqrt{k}} \biggr\Vert _{2,1}\cdot \biggl\Vert \frac{{T}_{l,l}-\mathrm{E}{T}_{l,l}-({T}_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta \delta_{l}\sqrt{l}} \biggr\Vert _{2,1}. \end{aligned}$$
$$\begin{aligned} \biggl\Vert \frac{{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta \delta_{k}\sqrt{k}} \biggr\Vert _{2,1} \leq& \frac{r}{r-2} \biggl\Vert \frac {{T}_{k,k}-\mathrm{E}{T}_{k,k}}{\beta\delta_{k}\sqrt{k}} \biggr\Vert _{r} \\ =& \frac{r}{r-2}\frac{1}{\beta\delta_{k}\sqrt{k}}\bigl(\mathrm{E} \vert {T}_{k,k}-\mathrm{E} {T}_{k,k} \vert ^{r} \bigr)^{\frac{1}{r}} \\ \leq&\frac{C}{\sqrt{k}}\Biggl(\sum_{i=1}^{k} \mathrm{E}|\bar {Y}_{ki}|^{r}+\Biggl(\sum _{i=1}^{k} \mathrm{E}\bar{Y}_{ki}^{2} \Biggr)^{r/2}\Biggr)^{1/r} \\ \leq&\frac{C}{\sqrt {k}} \bigl(k+k^{r/2}\bigr)^{1/r}\leq C, \end{aligned}$$
$$\biggl\Vert \frac{{T}_{l,l}-\mathrm{E}{T}_{l,l}-({T}_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta \delta_{l}\sqrt{l}} \biggr\Vert _{2,1}\leq C. $$
$$ I_{2}\leq C\rho^{-}(k). $$
(9)
Secondly, we prove (5); for \(\forall k \geq1\), \(\eta_{k}=f({\bar {V}_{k,1}^{2}}/({k\delta_{k,1}^{2}}))-\mathrm{E}(f({\bar {V}_{k,1}^{2}}/({k\delta_{k,1}^{2}})))\), we have
by the property of f, we know
$$\begin{aligned} \vert \mathrm{E}\eta_{k}\eta_{l} \vert =& \biggl\vert \operatorname{Cov}\biggl(f\biggl(\frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr),f\biggl( \frac{\bar{V}_{l,1}^{2}}{l\delta _{l,1}^{2}}\biggr) \biggr) \biggr\vert \\ \leq& \biggl\vert \operatorname{Cov}\biggl(f\biggl(\frac{\bar{V}_{k,1}^{2}}{k\delta _{k,1}^{2}} \biggr),f\biggl(\frac{\bar{V}_{l,1}^{2}}{l\delta_{l,1}^{2}}\biggr) - f \biggl(\frac{\sum_{i=2k+1}^{l}\bar{Y}_{l,i}^{2}I({Y}_{i}\geq 0)}{l\delta_{l,1}^{2}}\biggr) \biggr) \biggr\vert \\ &{}+ \biggl\vert \operatorname{Cov}\biggl(f\biggl( \frac{\bar{V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr),f \biggl(\frac{\sum_{i=2k+1}^{l}\bar{Y}_{l,i}^{2}I({Y}_{i}\geq0)}{l\delta _{l,1}^{2}}\biggr)\biggr) \biggr\vert \\ =&J_{1}+J_{2}, \end{aligned}$$
(10)
$$ J_{1}\leq C\Biggl(\mathrm{E}\Biggl(\sum_{i=1}^{2k} \bar{Y}_{ki}^{2}\mathrm{I}({Y}_{i}\geq 0)\Biggr)\Big/l \Biggr)\leq C\biggl(\frac{k}{l}\biggr). $$
(11)
Now we estimate \(J_{2}\),
and similarly \(\operatorname{Var}(\sum_{i=2k+1}^{l}\bar{Y}_{li}^{2}\mathrm{I}({Y}_{i}\geq0)/ (l\delta_{l,1}^{2}))\leq C\). On the other hand, we have
similarly
Thus, by Lemma 2.3, we have
hence, combining with (11) and (12), (3) holds, and by Lemma 2.4, (5) holds. □
$$\begin{aligned} \operatorname{Var} \biggl(\frac{\bar{V}_{k,1}^{2}}{k\delta _{k,1}^{2}} \biggr) =& \operatorname{Var} \biggl(\frac{\sum_{i=1}^{k}\bar {Y}_{ki}^{2}\mathrm{I}({Y}_{i}\geq0)}{k\delta_{k,1}^{2}} \biggr) \\ \leq& \frac{C}{k^{2}} \mathrm{E}\Biggl(\sum_{i=1}^{k} \bar{Y}_{ki}^{2}\mathrm{I}({Y}_{i}\geq0) \Biggr)^{2} \\ =&\frac{C}{k^{2}} \mathrm{E} \Biggl(\sum_{i=1}^{k} \bar{Y}_{ki}^{2}\mathrm{I}({Y}_{i}\geq0)-\mathrm{E} \Biggl(\sum_{i=1}^{k}\bar{Y}_{ki}^{2} \mathrm{I}({Y}_{i}\geq0)\Biggr) +\mathrm{E}\Biggl(\sum _{i=1}^{k}\bar{Y}_{ki}^{2} \mathrm{I}({Y}_{i}\geq0)\Biggr) \Biggr)^{2} \\ \leq& \frac{C}{k^{2}} \mathrm{E} \Biggl(\sum_{i=1}^{k} \bigl(\bar {Y}_{ki}^{2}\mathrm{I}({Y}_{i}\geq0)- \mathrm{E}\bigl(\bar{Y}_{ki}^{2}\mathrm{I}({Y}_{i} \geq0)\bigr)\bigr) \Biggr)^{2} \\ &{}+\frac{C}{k^{2}} \Biggl(\sum _{i=1}^{k}\mathrm{E}\bigl(\bar{Y}_{ki}^{2} \mathrm{I}({Y}_{i}\geq0)\bigr)\Biggr)^{2} \\ \leq&\frac{C}{k^{2}}\sum_{i=1}^{k} \mathrm{E}\bar{Y}_{ki}^{4}\mathrm{I}({Y}_{i} \geq0)+\frac{C}{k^{2}}\bigl(k\mathrm{E}\bigl(\bar{Y}_{k1}^{2} \mathrm{I}({Y}_{1}\geq0)\bigr)\bigr)^{2} \\ \leq&\frac{C}{k^{2}}\sum_{i=1}^{k} \mathrm{E}k(Y_{i} )^{2}\leq C, \end{aligned}$$
$$\begin{aligned} \biggl\Vert \frac{\bar{V}_{k,1}^{2}}{k\delta _{k,1}^{2}} \biggr\Vert _{2,1} \leq& \frac{r}{r-2}\cdot\frac{C}{k}\bigl(\mathrm{E} \bigl\vert \bar {V}_{k,1}^{2} \bigr\vert ^{r} \bigr)^{1/r} \\ \leq& \frac{C}{k}\Biggl(\mathrm{E} \Biggl\vert \sum _{i=1}^{k}\bigl(\bar {Y}_{ki}^{2} \mathrm{I}({Y}_{i}\geq0)- \mathrm{E}\bigl(\bar{Y}_{ki}^{2} \mathrm{I}({Y}_{i}\geq0)\bigr)\bigr) \Biggr\vert ^{r} + \Biggl\vert \sum_{i=1}^{k} \mathrm{E}\bigl( \bar{Y}_{ki}^{2}\mathrm{I}({Y}_{i}\geq 0)\bigr) \Biggr\vert ^{r}\Biggr)^{1/r} \\ \leq&\frac{C}{k}\Biggl(\sum_{i=1}^{k} \mathrm{E} \bigl\vert \bigl(\bar {Y}_{ki}^{2} \mathrm{I}({Y}_{i}\geq0)- \mathrm{E}\bigl(\bar{Y}_{ki}^{2} \mathrm{I}({Y}_{i}\geq0)\bigr)\bigr) \bigr\vert ^{r} \\ &{}+ \Biggl( \sum_{i=1}^{k}\mathrm{E} \bigl( \bar{Y}_{ki}^{2}\mathrm{I}({Y}_{i}\geq0)- \mathrm{E}\bigl(\bar{Y}_{ki}^{2}\mathrm{I}({Y}_{i} \geq0)\bigr)\bigr)^{2}\Biggr)^{r/2}\Biggr)^{1/r} \\ &{}+\frac{C}{k} \Biggl\vert \sum_{i=1}^{k} \mathrm{E}\bigl(\bar{Y}_{ki}^{2}\mathrm{I}({Y}_{i} \geq0)\bigr) \Biggr\vert \\ \leq& \frac{C}{k}\Biggl(\sum_{i=1}^{k} \mathrm{E} \bigl\vert \bar {Y}_{ki}^{2} \mathrm{I}({Y}_{i}\geq0) \bigr\vert ^{r} + \Biggl(\sum _{i=1}^{k} \mathrm{E} \bigl\vert \bar{Y}_{ki}^{2}\mathrm{I}({Y}_{i}\geq0) \bigr\vert ^{2}\Biggr)^{r/2}\Biggr)^{1/r} \\ &{}+ \frac{C}{k} \bigl\vert k \mathrm{E}\bigl(\bar{Y}_{k1}^{2} \mathrm{I}({Y}_{1}\geq0)\bigr) \bigr\vert \\ \leq&\frac{C}{k}\Biggl(\sum_{i=1}^{k} \mathrm{E} \vert \sqrt {k} {Y}_{i} \vert ^{r}+\Biggl( \sum_{i=1}^{k} \mathrm{E} \vert \sqrt {k} {Y}_{i} \vert ^{2}\Biggr)^{r/2} \Biggr)^{1/r} +C_{1} \\ \leq& \frac{C}{k}\bigl(k^{1+{r/2}}+k^{r} \bigr)^{1/r}+C_{1}\leq C, \end{aligned}$$
$$\Biggl\Vert \sum_{i=2k+1}^{l} \bar{Y}_{li}^{2}\mathrm{I}({Y}_{i}\geq0)/ \bigl(l \delta_{l,1}^{2}\bigr) \Biggr\Vert _{2,1}\leq C. $$
$$\begin{aligned} J_{2} \leq& C\biggl\{ -\operatorname{Cov}\biggl(\frac{\bar{V}_{k,1}^{2}}{k\delta _{k,1}^{2}}, \frac{\sum_{i=2k+1}^{l}\bar{Y}_{li}^{2}\mathrm{I}({Y}_{i}\geq 0)}{l\delta_{l,1}^{2}}\biggr) \\ &{}+8\rho^{-}\biggl( \frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}},\frac{\sum_{i=2k+1}^{l}\bar {Y}_{li}^{2}\mathrm{I}({Y}_{i}\geq0)}{l\delta_{l,1}^{2}} \biggr) \cdot \biggl\Vert \frac{\bar{V}_{k,1}^{2}}{k\delta_{k,1}^{2}} \biggr\Vert _{2,1}\cdot \biggl\Vert \frac{\sum_{i=2k+1}^{l}\bar{Y}_{li}^{2}\mathrm{I}({Y}_{i}\geq0)}{l\delta _{l,1}^{2}} \biggr\Vert _{2,1}\biggr\} \\ \leq& C\biggl\{ \rho^{-}(k) \biggl(\operatorname{Var}\biggl( \frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr)\biggr)^{1/2}\cdot \operatorname{Var}\biggl( \frac {\sum_{i=2k+1}^{l}\bar{Y}_{li}^{2}\mathrm{I}({Y}_{i}\geq0)}{l\delta _{l,1}^{2}}\biggr)^{1/2} \\ &{}+\rho^{-}(k)\cdot \biggl\Vert \frac{\bar{V}_{k,1}^{2}}{k\delta _{k,1}^{2}} \biggr\Vert _{2,1}\cdot \biggl\Vert \frac{\sum_{i=2k+1}^{l}\bar {Y}_{li}^{2}\mathrm{I}({Y}_{i}\geq0)}{l\delta_{l,1}^{2}} \biggr\Vert _{2,1} \biggr\} \\ \leq&C\rho^{-}(k), \end{aligned}$$
(12)
3 Proof of Theorem 1
Let \(C_{k,i}= \frac{S_{k,i}}{(k-1)\mu}\), hence, (2) is equivalent to
So we only need to prove (13), for a fixed k, \(1\leq k\leq n\) and \(\forall\varepsilon>0\); we have
therefore, by Theorem 1.5.2 in [14], we have
on the unanimous establishment of i.
$$ \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1}^{n} d_{k}\mathrm{I}\Biggl(\frac{\mu}{\beta V_{k}}\sum _{i=1}^{k}\log C_{k,i}\leq{x}\Biggr)= \Phi(x) \quad \mbox{a.s.} $$
(13)
$$\lim_{k\rightarrow\infty}P\Biggl\{ \bigcup_{m=k}^{\infty} \biggl( \biggl\vert \frac {X_{i}}{m} \biggr\vert \geq\varepsilon\biggr) \Biggr\} =\lim_{k\rightarrow\infty}P\biggl\{ \biggl\vert \frac{X_{i}}{k} \biggr\vert \geq\varepsilon \biggr\} =\lim_{k\rightarrow\infty}P\bigl\{ | X_{1}|\geq \varepsilon k\bigr\} =0, $$
$$\frac{X_{i}}{k}\rightarrow0\quad \mbox{a.s. }k\rightarrow\infty, $$
By Lemma 2.1, for some \(\frac{4}{3}< p<2 \), and enough large k, we have
by \(\log(1+x)=x+O(x^{2})\), \(x\rightarrow0\), we get
and then, for \(\delta>0\) and every ω, there exists \(k_{0}=k_{0}(\omega,\delta,x)\); when \(k>k_{0}\), we have
under the condition \(|X_{i}-\mu|\leq\sqrt{k}\), \(1\leq i\leq k\), we have
furthermore, by (14) and (15), for any given \(0<\varepsilon<1\), \(\delta >0\), when \(k>k_{0}\), we obtain
Therefore, to prove (13), for any \(0<\varepsilon<1\), \(\delta_{1}>0\), it suffices to prove
$$\begin{aligned} \sup_{1\leq i\leq k} \vert C_{k,i}-1 \vert \leq& \biggl\vert \frac{\sum_{j=1}^{k}(X_{j}-\mu)}{(k-1)\mu} \biggr\vert +\sup_{1\leq i\leq k} \biggl\vert \frac{X_{i}}{(k-1)\mu} \biggr\vert +\frac {1}{k-1} \\ \leq& \biggl\vert \frac{S_{k}-k\mu}{k^{\frac{1}{p}}}\cdot\frac{k^{\frac {1}{p}}}{(k-1)\mu} \biggr\vert \leq Ck^{\frac{1}{p}-1}, \end{aligned}$$
$$\begin{aligned} &\Biggl\vert \frac{\mu}{\beta\delta_{k}\sqrt{(1\pm\varepsilon)k}}\sum_{i=1}^{k} \ln{C}_{k,i}-\frac{\mu}{\beta\delta_{k}\sqrt{(1\pm\varepsilon )k}}\sum_{i=1}^{k}(C_{k,i}-1) \Biggr\vert \\ &\quad \leq\frac{{C}\mu}{\beta\delta_{k}\sqrt{(1\pm\varepsilon)k}}\sum_{i=1}^{k}(C_{k,i}-1)^{2} \\ &\quad \leq\frac{C}{\sqrt{k}}k^{\frac{2}{p}-1}\rightarrow0\quad \mbox{a.s.}, k \rightarrow \infty, \end{aligned}$$
$$\begin{aligned} &\mathrm{I}\Biggl\{ \frac{\mu}{\beta\delta_{k}\sqrt{(1\pm\varepsilon)k}}\sum_{i=1}^{k}(C_{k,i}-1) \leq{x}-\delta\Biggr\} \\ &\quad \leq\mathrm{I}\Biggl\{ \frac{\mu}{\beta\delta_{k}\sqrt{(1\pm\varepsilon )k}}\sum _{i=1}^{k}\log{C}_{k,i}\leq{x}\Biggr\} \\ &\quad \leq\mathrm{I}\Biggl\{ \frac{\mu}{\beta\delta_{k}\sqrt{(1\pm\varepsilon )k}}\sum_{i=1}^{k}(C_{k,i}-1) \leq{x}+\delta\Biggr\} , \end{aligned}$$
(14)
$$ \mu\sum_{i=1}^{k}(C_{k,i}-1)=\sum _{i=1}^{k}\frac{S_{k,i}-(k-1)\mu }{k-1}=\sum _{i=1}^{k}Y_{i}=\sum _{i=1}^{k}\bar{Y}_{ki}={T}_{k,i}, $$
(15)
$$\begin{aligned}& \mathrm{I}\Biggl(\frac{\mu}{\beta V_{k}}\sum_{i=1}^{k} \log C_{k,i}\leq x\Biggr) \\& \quad \leq\mathrm{I}\biggl(\frac{{T}_{k,i}}{\delta_{k}\beta\sqrt{k(1+\varepsilon )}}\leq x+ \delta\biggr) +\mathrm{I}\bigl(\bar{V}_{k}^{2}>(1+ \varepsilon)k\delta_{k}^{2}\bigr) \\& \qquad {}+\mathrm{I}\Biggl(\bigcup _{i=1}^{k}\bigl( \vert X_{i}-\mu \vert >\sqrt{k}\bigr)\Biggr),\quad x\geq0, \\& \mathrm{I}\Biggl(\frac{\mu}{\beta V_{k}}\sum_{i=1}^{k} \log C_{k,i}\leq x\Biggr) \\& \quad \leq\mathrm{I}\biggl(\frac{{T}_{k,i}}{\delta_{k}\beta\sqrt{k(1-\varepsilon )}} \leq x+ \delta\biggr) +\mathrm{I}\bigl(\bar{V}_{k}^{2}< (1- \varepsilon)k\delta_{k}^{2}\bigr) \\& \qquad {}+\mathrm{I}\Biggl(\bigcup _{i=1}^{k}\bigl( \vert X_{i}-\mu \vert >\sqrt{k}\bigr)\Biggr),\quad x< 0, \\& \mathrm{I}\Biggl(\frac{\mu}{\beta V_{k}}\sum_{i=1}^{k} \log C_{k,i}\leq x\Biggr) \\& \quad \geq\mathrm{I}\biggl(\frac{{T}_{k,i}}{\delta_{k}\beta\sqrt{k(1-\varepsilon )}}\leq x- \delta\biggr) -\mathrm{I}\bigl(\bar{V}_{k}^{2}< (1- \varepsilon)k\delta_{k}^{2}\bigr) \\& \qquad {}-\mathrm{I}\Biggl(\bigcup _{i=1}^{k}\bigl( \vert X_{i}-\mu \vert >\sqrt{k}\bigr)\Biggr),\quad x\geq0, \\& \mathrm{I}\Biggl(\frac{\mu}{\beta V_{k}}\sum_{i=1}^{k} \log C_{k,i}\leq x\Biggr) \\& \quad \geq\mathrm{I}\biggl(\frac{{T}_{k,i}}{\delta_{k}\beta\sqrt{k(1+\varepsilon )}}\leq x- \delta\biggr) -\mathrm{I}\bigl(\bar{V}_{k}^{2}>(1+ \varepsilon)k\delta_{k}^{2}\bigr) \\& \qquad {}-\mathrm{I}\Biggl(\bigcup _{i=1}^{k}\bigl( \vert X_{i}-\mu \vert >\sqrt{k}\bigr)\Biggr),\quad x< 0. \end{aligned}$$
$$\begin{aligned}& \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{I} \biggl( \frac{{T}_{k,i}}{\beta\delta_{k}\sqrt{k}}\leq\sqrt{1\pm\varepsilon }x\pm \delta_{1} \biggr)=\Phi(\sqrt{1\pm\varepsilon}x\pm\delta_{1}) \quad \mbox{a.s.} , \end{aligned}$$
(16)
$$\begin{aligned}& \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{I}\Biggl(\bigcup_{i=1}^{k}\bigl( \vert X_{i}-\mu \vert >\sqrt{k}\bigr)\Biggr)=0\quad \mbox{a.s.} , \end{aligned}$$
(17)
$$\begin{aligned}& \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{I} \bigl(\bar{V}_{k}^{2}>(1+\varepsilon)k \delta_{k}^{2}\bigr)=0 \quad \mbox{a.s.} , \end{aligned}$$
(18)
$$\begin{aligned}& \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{I} \bigl(\bar{V}_{k}^{2}< (1-\varepsilon)k \delta_{k}^{2}\bigr)=0 \quad \mbox{a.s.} \end{aligned}$$
(19)
Firstly, we prove (16), by \(\mathrm{E}(Y^{2})<\infty\), we know \(\lim_{x\rightarrow\infty}x^{2}P(|Y|>x)=0\), and by \(\mathrm{E}(Y)=0\), it follows that
so, combining with \(\delta_{k}^{2}\rightarrow\mathrm{E}(Y^{2})<\infty\), for any \(\alpha>0\), when \(k\rightarrow\infty\), we have
thus, by (4), we get
letting \(\alpha\rightarrow0\) in (20) and (21), (16) holds.
$$\begin{aligned} \bigl\vert \mathrm{E}({T}_{k,i}) \bigr\vert =& \Biggl\vert \mathrm{E}\Biggl(\sum_{i=1}^{k}\bar {Y}_{ki}\Biggr) \Biggr\vert = \vert k\mathrm{E} \bar{Y}_{k1} \vert \\ \leq& k \bigl\vert \mathrm{E}\bigl(Y\mathrm{I}\bigl( \vert Y \vert > \sqrt{k}\bigr)\bigr) \bigr\vert +k^{\frac{3}{2}}\mathrm{E}\bigl(\mathrm{I} \bigl( \vert Y \vert >\sqrt{k}\bigr)\bigr) \\ \leq&\sqrt{k}\mathrm{E}\bigl(Y^{2}\mathrm{I}\bigl( \vert Y \vert >\sqrt{k}\bigr)\bigr)+k^{\frac {3}{2}}P\bigl( \vert Y \vert >\sqrt{k} \bigr)=o(\sqrt{k}), \end{aligned}$$
$$\begin{aligned} &\mathrm{I} \biggl(\frac{{T}_{k,i}-E{T}_{k,i}}{\beta\delta _{k}\sqrt{k}}\leq\sqrt{1\pm\varepsilon} x\pm \delta_{1}-\alpha \biggr) \\ &\quad \leq \mathrm{I} \biggl(\frac{{T}_{k,i}}{\beta\delta_{k}\sqrt{k}} \leq\sqrt {1\pm\varepsilon}x\pm\delta_{1} \biggr) \\ &\quad \leq \mathrm{I} \biggl(\frac{{T}_{k,i}-E{T}_{k,i}}{\beta\delta_{k}\sqrt{k}}\leq \sqrt{1\pm\varepsilon}x\pm \delta_{1}+\alpha \biggr), \end{aligned}$$
$$\begin{aligned}& \begin{aligned}[b] &\lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum _{k=1}^{n}d_{k} \mathrm{I} \biggl( \frac{{T}_{k,i}}{\beta\delta_{k}\sqrt {k}}\leq\sqrt{1\pm\varepsilon}x\pm\delta_{1} \biggr) \\ &\quad \geq \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum _{k=1}^{n}d_{k} \mathrm{I} \biggl( \frac{{T}_{k,i}-E{T}_{k,i}}{\beta\delta_{k}\sqrt{k}}\leq \sqrt{1\pm\varepsilon}x\pm\delta_{1}-\alpha \biggr) \\ &\quad \rightarrow\Phi(\sqrt{1\pm\varepsilon}x\pm\delta_{1}-\alpha), \end{aligned} \end{aligned}$$
(20)
$$\begin{aligned}& \begin{aligned}[b] &\lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum _{k=1}^{n}d_{k} \mathrm{I} \biggl( \frac{{T}_{k,i}}{\beta\delta_{k}\sqrt {k}}\leq\sqrt{1\pm\varepsilon}x\pm\delta_{1} \biggr) \\ &\quad \leq\lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum _{k=1}^{n}d_{k}\mathrm{I} \biggl( \frac{{T}_{k,i}-E{T}_{k,i}}{\beta\delta_{k}\sqrt{k}}\leq\sqrt {1\pm\varepsilon} x\pm\delta_{1}+\alpha \biggr) \\ &\quad \rightarrow \Phi(\sqrt{1\pm\varepsilon}x\pm\delta_{1}+\alpha)\quad \mbox{a.s.}, \end{aligned} \end{aligned}$$
(21)
Now, we prove (17); by \(\mathrm{E}(Y^{2})<\infty\), we know \(\lim_{x\rightarrow\infty}x^{2}P(|Y|>x)=0\), such that
by the Toeplitz lemma, we get
hence, to prove (17), it suffices to prove
writing
for every \(0\leq2k< l\), so by the definition of \(\rho^{-}\)-mixing sequence, we have
so by Lemma 2.4, (23) holds. And combining with (22), we know that (17) holds.
$$\mathrm{E}\mathrm{I}\Biggl(\bigcup_{i=1}^{k} \bigl( \vert Y_{i} \vert >\sqrt{k}\bigr)\Biggr)\leq\sum _{i=1}^{k}P\bigl( \vert Y_{i} \vert > \sqrt{k}\bigr)\leq kP\bigl( \vert Y \vert >\sqrt{k}\bigr)\rightarrow 0,\quad k \rightarrow\infty, $$
$$ \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{E}\mathrm{I}\Biggl(\bigcup_{i=1}^{k} \bigl( \vert Y_{i} \vert >\sqrt{k}\bigr)\Biggr)\rightarrow0\quad \mbox{a.s.}, $$
(22)
$$ \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \Biggl(\mathrm{I}\Biggl(\bigcup_{i=1}^{k} \bigl( \vert Y_{i} \vert >\sqrt{k}\bigr)\Biggr)-\mathrm{E} \Biggl[ \mathrm{I}\Biggl(\bigcup_{i=1}^{k}\bigl( \vert Y_{i} \vert >\sqrt{k}\bigr)\Biggr) \Biggr] \Biggr) \rightarrow0\quad \mbox{a.s.}, $$
(23)
$$\mathscr {Z}_{k}= \mathrm{I}\Biggl(\bigcup _{i=1}^{k}\bigl( \vert Y_{i} \vert > \sqrt{k}\bigr)\Biggr)-\mathrm{E} \Biggl[\mathrm{I}\Biggl(\bigcup _{i=1}^{k}\bigl( \vert Y_{i} \vert > \sqrt{k}\bigr)\Biggr) \Biggr], $$
$$\begin{aligned} \mathrm{E} \vert \mathscr {Z}_{k}\mathscr {Z}_{l} \vert =& \Biggl\vert \operatorname{Cov}\Biggl(\mathrm{I}\Biggl(\bigcup _{i=1}^{k}\bigl( \vert Y_{i} \vert > \sqrt{k}\bigr)\Biggr),\mathrm{I}\Biggl(\bigcup_{i=1}^{l} \bigl( \vert Y_{i} \vert >\sqrt{l}\bigr)\Biggr) \Biggr) \Biggr\vert \\ \leq& \Biggl\vert \operatorname{Cov}\Biggl(\mathrm{I}\Biggl(\bigcup _{i=1}^{k}\bigl( \vert Y_{i} \vert > \sqrt {k}\bigr)\Biggr),\mathrm{I}\Biggl(\bigcup_{i=1}^{l} \bigl( \vert Y_{i} \vert >\sqrt{l}\bigr)\Biggr)-\mathrm{I}\Biggl( \bigcup_{i=2k+1}^{l}\bigl( \vert Y_{i} \vert >\sqrt{l}\bigr)\Biggr) \Biggr) \Biggr\vert \\ &{}+ \Biggl\vert \operatorname{Cov}\Biggl(\mathrm{I}\Biggl(\bigcup _{i=1}^{k}\bigl( \vert Y_{i} \vert > \sqrt {k}\bigr)\Biggr),\mathrm{I}\Biggl(\bigcup_{i=2k+1}^{l} \bigl( \vert Y_{i} \vert >\sqrt{l}\bigr)\Biggr) \Biggr) \Biggr\vert \\ \leq&\mathrm{E} \Biggl\vert \mathrm{I}\Biggl(\bigcup _{i=1}^{l}\bigl( \vert Y_{i} \vert > \sqrt{l}\bigr)\Biggr)-\mathrm{I}\Biggl(\bigcup_{i=2k+1}^{l} \bigl( \vert Y_{i} \vert >\sqrt{l}\bigr)\Biggr) \Biggr\vert \\ &{}+ \rho^{-}(k)\sqrt{\operatorname{Var}\Biggl(\mathrm{I} \Biggl(\bigcup_{i=1}^{k}\bigl( \vert Y_{i} \vert >\sqrt {k}\bigr)\Biggr)\Biggr)\operatorname{Var}\Biggl( \mathrm{I}\Biggl(\bigcup_{i=2k+1}^{l}\bigl( \vert Y_{i} \vert >\sqrt{l}\bigr)\Biggr)\Biggr)} \\ \leq&\mathrm{E}\Biggl[\mathrm{I}\Biggl(\bigcup_{i=1}^{2k} \bigl( \vert Y_{i} \vert >\sqrt{l}\bigr)\Biggr) \Biggr]+C \rho^{-}(k) \\ \leq&\sum_{i=1}^{k}P\bigl( \vert Y_{i} \vert >\sqrt{l}\bigr)+C\rho^{-}(k) \\ \leq& kP\bigl( \vert Y \vert >\sqrt{l}\bigr)+C\rho^{-}(k) \\ \leq& C\biggl(\frac{k}{l}+\rho^{-}(k)\biggr), \end{aligned}$$
Next, we prove (18); by \(\mathrm{E}(\bar{V}_{k}^{2})=k\delta_{k}^{2}\), \(\bar{V}_{k}^{2}=\bar{V}_{k,1}^{2}+\bar{V}_{k,2}^{2}\), \(\mathrm{E}(\bar {V}_{k,l}^{2})=k\delta_{k,l}^{2}\), and \(\delta_{k,1}^{2}\leq\delta_{k}^{2}\), \(l=1,2\), we have
therefore, by the arbitrariness of \(\varepsilon>0\), to prove (18), it suffices to prove
when \(l=1\), for given \(\varepsilon>0\), let f be a bounded function with bounded continuous derivative such that
under the condition
by the Markov inequality, and Lemma 2.2, we get
because \(\mathrm{E}(Y^{2})<\infty\) implies \(\lim_{x\rightarrow\infty }x^{2}P(|Y|>x)=0\), we have
thus, combining with (26),
Therefore, from (5), (25) and the Toeplitz lemma
hence, (24) holds for \(l=1\). Similarly, we can prove (24) for \(l=2\), so (18) is true. By similar methods used to prove (18), we can prove (19), this completes the proof of Theorem 1.
$$\begin{aligned} \mathrm{I}\bigl(\bar{V}_{k}^{2}>(1+\varepsilon)k \delta_{k}^{2}\bigr) =&\mathrm{I}\bigl(\bar {V}_{k}^{2}-\mathrm{E}\bigl(\bar{V}_{k}^{2} \bigr)>\varepsilon k\delta_{k}^{2}\bigr) \\ \leq& \mathrm{I}\bigl(\bar{V}_{k,1}^{2}-\mathrm{E}\bigl(\bar {V}_{k,1}^{2}\bigr)>\varepsilon k\delta_{k}^{2}/2 \bigr)+\mathrm{I}\bigl(\bar {V}_{k,2}^{2}-\mathrm{E}\bigl( \bar{V}_{k,2}^{2}\bigr)>\varepsilon k\delta _{k}^{2}/2\bigr) \\ \leq& \mathrm{I}\biggl(\bar{V}_{k,1}^{2}> \biggl(1+ \frac{\varepsilon}{2}\biggr)k\delta _{k,1}^{2} \biggr)+ \mathrm{I} \biggl(\bar{V}_{k,2}^{2}> \biggl(1+\frac{\varepsilon}{2}\biggr)k \delta_{k,2}^{2}\biggr), \end{aligned}$$
$$ \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{I} \biggl(\bar{V}_{k,l}^{2}>\biggl(1+\frac{\varepsilon}{2} \biggr)k\delta_{k,l}^{2}\biggr)=0\quad \mbox{a.s. } l=1,2, $$
(24)
$$ \mathrm{I}(x>1+\varepsilon)\leq f(x)\leq\mathrm{I}\biggl(x>1+\frac{\varepsilon }{2} \biggr), $$
(25)
$$\mathrm{E}\bigl(\bar{V}_{k,1}^{2}\bigr)=k \delta_{k,1}^{2},\qquad \mathrm{E}\bigl(Y^{2}\bigr)< \infty,\qquad \mathrm{E}\bigl(Y^{2}\mathrm{I}(Y\geq0)\bigr)>0, $$
$$\begin{aligned} &P\biggl(\bar{V}_{k,1}^{2}>\biggl(1+\frac{\varepsilon}{2}\biggr)k \delta _{k,1}^{2}\biggr) \\ &\quad =P\biggl(\bar{V}_{k,1}^{2}- \mathrm{E}\bigl(\bar{V}_{k,1}^{2}\bigr)>\frac {\varepsilon}{2} k \delta_{k,1}^{2}\biggr) \\ &\quad \leq C\frac{\mathrm{E}(\bar{V}_{k,1}^{2}-\mathrm{E}(\bar {V}_{k,1}^{2}))^{2}}{k^{2}} \leq C\frac{\sum_{i=1}^{k}\mathrm{E}(\bar{Y}_{ki}^{2}\mathrm{I}(\bar {Y}_{ki}\geq0))^{2}}{k^{2}} \\ &\quad \leq C\frac{\mathrm{E}\bar{Y}_{k1}^{4}\mathrm{I}(\bar{Y}_{k1}\geq0)}{k} \leq C\frac{\mathrm{E}Y^{4}\mathrm{I}(0\leq Y\leq\sqrt{k})+k^{2}P(Y>\sqrt{k})}{k}, \end{aligned}$$
(26)
$$\begin{aligned} \mathrm{E}Y^{4}\mathrm{I}(0\leq Y\leq\sqrt{k}) =& \int _{0}^{\infty}P\bigl( \vert Y \vert \mathrm{I}(0 \leq Y\leq\sqrt{k})\geq t \bigr)4t^{3}\,dt \\ \leq&C \int_{0}^{\sqrt{k}}P\bigl( \vert Y \vert \geq t \bigr)t^{3}\,dt \\ =& \int_{0}^{\sqrt{k}}o(1)t\, dt =o(1)k, \end{aligned}$$
$$P\biggl(\bar{V}_{k,1}^{2}>\biggl(1+\frac{\varepsilon}{2}\biggr)k \delta _{k,1}^{2}\biggr)\rightarrow0 ,\quad k\rightarrow\infty. $$
$$\begin{aligned} 0 \leq& \frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{I} \biggl(\bar{V}_{k,1}^{2}>\biggl(1+\frac{\varepsilon}{2} \biggr)k\delta_{k,1}^{2}\biggr) \\ \leq&\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k}f \biggl(\frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr) \\ =&\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{E} \biggl(f\biggl(\frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr) \biggr) +\frac{1}{D_{n}} \sum_{k=1}^{n}d_{k} \biggl(f \biggl(\frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr)-\mathrm{E}\biggl(f\biggl(\frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}} \biggr)\biggr) \biggr) \\ \leq&\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \mathrm{E}\biggl(\mathrm{I} \biggl(\bar {V}_{k,1}^{2}>\biggl(1+ \frac{\varepsilon}{2}\biggr)k\delta_{k,1}^{2}\biggr)\biggr) + \frac{1}{D_{n}}\sum_{k=1}^{n}d_{k} \biggl(f\biggl(\frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr)-\mathrm{E}\biggl(f\biggl( \frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr)\biggr) \biggr) \\ =&\frac{1}{D_{n}}\sum_{k=1}^{n}d_{k}P \biggl(\bar{V}_{k,1}^{2}>\biggl(1+\frac {\varepsilon}{2}\biggr)k \delta_{k,1}^{2}\biggr) +\frac{1}{D_{n}}\sum _{k=1}^{n}d_{k} \biggl(f\biggl( \frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr)-\mathrm{E}\biggl(f\biggl(\frac{\bar {V}_{k,1}^{2}}{k\delta_{k,1}^{2}}\biggr) \biggr) \biggr) \\ \rightarrow&0 \quad \mbox{a.s.}, k\rightarrow\infty, \end{aligned}$$
Authors’ information
XiLi Tan, Professor, Doctor, working in the field of probability and statistics. Wei Liu, Master, working in the field of probability and statistics.
Competing interests
The authors declare that there is no conflict of interest regarding the publication of this paper. We confirm that the received funding mentioned in the “Acknowledgment” section did not lead to any conflict of interests regarding the publication of this manuscript. We declare that we do not have any commercial or associated interest that represents a conflict of interest in connection with the work submitted.
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