The law of iterated logarithm for a class of random variables satisfying Rosenthal type inequality

1.   Introduction and main results

We first introduce the definition of the Rosenthal type maximal inequality, which is one of the most interesting inequalities in probability theory and mathematical statistics. Suppose that $\{Y_n, n\geq 1\}$ is a sequence of random variables satisfying $E|Y_i|^r < \infty$ for $r\geq 2$, then there exists a positive constant $C(r)$ depending only on $r$ such that

(1.1) can be satisfied by many dependent or mixing sequences. Peligrad ^[1], Zhou ^[2], Wang and Lu ^[3], Utev and Peligrad ^[4] established the above inequality for $\rho-$mixing sequence, $\varphi-$mixing sequence, $\rho^-$mixing sequence and $\widetilde{\rho}-$mixing sequence, respectively. We also refer to Shao ^[5], Stoica ^[6], Shen ^[7], Yuan and An ^[8], Shen et al. ^[9] and Merlev$\acute{e}$de and Peligrad ^[10] for negatively associated sequence (NA), martingale difference sequence, extend negatively dependent sequence (END), asymptotically almost negatively associated random sequence (AANA), negatively superadditive dependent (NSD), stationary processes, respectively.

The law of iterated logarithm (LIL, for short) is an important aspect in probability theory because it can describe the precise convergence rates. Petrov ^[11] established the following LIL for independent random variables.

Theorem A. Let $\{X_{n}, n\geq1\}$ be independent random variables sequences with $EX_n = 0$, $\sigma_n^2 = EX_n^2 < \infty$, $B_n' = \sum_{k = 1}^n \sigma_k^2$, $S_n = \sum_{k = 1}^nX_{k}$. If the following assumptions are satisfied:

(i) $B_n'\to \infty$, when $n\to\infty$,

(ii) $B_{n+1}'/{B_n'}\to 1$, when $n\to\infty$,

(iii) $\Delta_n = \sup_{x}|P(S_n < x\sqrt{B_n'})-\Phi(x)| = O[(\log B_n')^{-1-\delta}]$, $\delta > 0$, here and in the sequel $\Phi(\cdot)$ is a standard normal distribution function, hold.

Then

Later on, Cai and Wu ^[12] established the following LIL for NA random variables.

Theorem B. Let $\{X_n, n\geq 1\}$ be NA random variables sequence with $EX_n = 0$ and $\sup_n EX_n^2(\log |X_n|)^{1+\delta} < \infty$ for some $\delta > 0$. Let $S_n = \sum_{k = 1}^nX_{k}$, $B_n = Var(S_n) > 0$, $B_n' = \sum_{k = 1}^n EX_k^2$. $\Delta_n = \sup_{x}|P(S_n < x\sqrt{B'_n})-\Phi(x)|$. If

(i) $B_n = O(n)$,

(ii) $B_{n+1}/{B_n}\to 1$, when $n\to\infty$,

(iii) $\Delta_n = O[(\log B_n)^{-1}]$,

(iv) $B_{n}/{B_n'}\to 1$, when $n\to\infty$,

hold, then

In this paper, the main purpose is to establish the law of the iterated logarithm for a class of random variables satisfying Rosenthal type maximal inequality with non-identical distributions. The following is the main result.

Theorem 1.1. Let $\{Y_n, n\geq 1\}$ be a sequence of random variables with $EY_n = 0$ and $\sup_nE|Y_n|^p < \infty$ for each $p > 2$ satisfying (1.1), denote $S_n = \sum_{k = 1}^nY_{k}$, $B_n = Var(S_n) > 0$. If

(i) $B_n = O(n)$, $B_{n+1}/{B_n}\to 1$, when $n\to\infty$,

(ii) $\Delta_{n, m} = \sup_{x}|P(S_{n+m}-S_m < x\sqrt{B_{n+m}-B_m})-\Phi(x)| = O[(\log (B_{n+m}-B_m))^{-1-\delta}]$ for some $\delta > 0$ and any $m\geq 0$, $n\geq 1$, where $S_0 = 0$, $B_0 = 0$,

hold, then

Corollary 1.2. Let $\{Y_n, n\geq 1\}$ be a strictly stationary sequence of random variables with $EY_1 = 0$ and $E|Y_1|^p < \infty$ for each $p > 2$ satisfying (1.1), denote $S_n = \sum_{k = 1}^nY_{k}$, $B_n = Var(S_n) > 0$. If

(i) $0 < \sigma^2 = :EY_1^2+\sum_{k = 1}^{\infty}EY_1Y_{1+k} < \infty$,

(ii) $\Delta_n = \sup_{x}|P(S_n < x\sqrt{B_n})-\Phi(x)| = O[(\log B_n)^{-1-\delta}]$ for some $\delta > 0$,

hold, then

Remark 1.3. The assumption (ii) in Theorem 1.1, that is the Berry-Esseen bounds, can be satisfied by many sequence, such as independent sequence, NA sequence with convergence rate $n^{-\alpha}$, $0 < \alpha\leq 1/2.$

Throughout the sequel, $C$ represents a positive constant although its value may change from one appearance to the next, $I\{A\}$ denotes the indicator function of the set $A$, $[x]$ denotes the integer part of $x$, $\log x = \ln{\max\{e, x\}}$.

2.   Proof

Some lemmas which will be useful to prove the main results are given firstly.

Lemma 2.1. (Wittmann ^[13]) Let $\{a_n\}$ be a sequence of strictly positive real numbers with $\lim_{n\to\infty}a_n = \infty$. Then for any $M > 1$, there exists a subsequence $\{n_k, k\geq 1\}\in {\mathbb{N}} = \{1, 2, \cdots\}$, such that

Lemma 2.2. Under the assumptions of Theorem 1.1, let $\{g(n)\}$ be a nondecreasing sequence of positive numbers and $\{n_k\}$ be a nondecreasing sequence of positive integers such that $\sum_{k = 1}^{\infty}\frac{1}{(\log n_k)^{1+\delta}} < \infty$. Then the following statements are equivalent

(A) $\sum_{k = 1}^{\infty}P(S_{n_k} > g(n_k)\sqrt{B_{n_k}}) < \infty$,

(B) $\sum_{k = 1}^{\infty}\frac{1}{g(n_k)}\exp\{-\frac{1}{2}g^2(n_k)\} < \infty.$

Proof. Noting $B_n = O(n)$,

Thanks to (2.1), condition (A) is equivalent to

If $g(n_k)\nrightarrow \infty$, it is easy to see that conditions (A) and (B) can not be satisfied. So we can assume that $g(n_k)\to \infty$, then noting that $\frac{1}{x}\varphi(x)\sim 1-\Phi(x)$ for $x$ large enough, where $\varphi(x)$ is the density function of the standard normal, one gets

Thus, the proof of this lemma is completed.

Lemma 2.3. Under the conditions of Theorem 1.1, one gets

Proof. For any $0 < \varepsilon < 1/3$, let $n_k = [e^{k^{\alpha}}], \; k\geq 1$ with $\max\{\frac{1}{(1+\varepsilon)^2}, \frac{1}{1+\delta}\} < \alpha < 1$ and $g(n_k) = (1+\varepsilon)(2\log\log B_{n_k})^{1/2}$. Noting that $B_n = O(n)$, one can get

and

Then by Lemma 2.2, one can obtain

By Broel-Cantelli lemma and the arbitrariness of $\varepsilon$, we have

For given $\alpha$, choose $p > 2$ such that $p(1-\alpha) > 2$, then by (1.1), $B_n = O(n)$ and $\sup_nE|Y_n|^p < \infty$, we get

Thus by Broel-Cantelli lemma and the arbitrariness of $\varepsilon$, one has

Thanks to (2.3) and (2.4) and $B_{n_{k+1}}/{B_{n_k}}\to 1$, we have

thus, the proof of Lemma 2.3 is completed.

Lemma 2.4. Under the conditions of Theorem 1.1, one gets

Proof. Noting that $B_n = O(n)$, $B_{n+1}/{B_n}\to 1$, by Lemma 2.1, for any $\tau > 0$, there exists a nondecreasing sequence of positive integers $\{n'_k, k\geq 1\}$, such that for $k\to\infty$, we have

Let

From (1.3), it is easy to check that

For given $0 < \varepsilon < 1$, one can choose $0 < \theta < 1$ and $\tau > 0$, such that

Let $g(n'_k) = (1+\varepsilon)(2\log\log B_{n'_k})^{1/2}$. Noting that $B_n = O(n)$, one can get

and

Then by Lemma 2.2, one can obtain

By Broel-Cantelli lemma and $0 < \varepsilon < 1$, we have

In order to prove (2.5), it is sufficient to show that

Noting $(1-\theta)\tau^{1/2}(1+\tau)^{-1/2}-2(1+\tau)^{-1/2} > 1-\varepsilon,$ then by (2.6) and (2.8) and $P(AB)\geq P(A)-P(\overline{B})$, it is easy to prove

Thus by (2.10), in order to prove (2.9), it suffices to prove

Noting $\Delta_{n, m} = \sup_{x}|P(S_{n+m}-S_m < x\sqrt{B_{n+m}-B_m})-\Phi(x)| = O[(\log (B_{n+m}-B_m))^{-1-\delta}]$ and $\frac{1}{x}\varphi(x)\leq 1-\Phi(x)$ for $x\geq 1$, where $\varphi(x)$ is the density function of the standard normal random variables, recall $\psi(n'_k) = (2(B_{n'_k}-B_{n'_{k-1}})\log\log (B_{n'_k}-B_{n'_{k-1}}))^{1/2}$ and $B_{n'_{k}}\sim (1+\tau)^k$, one can deduce

Hence, by the generalized Borel-Cantelli lemma (see, e.g., Kochen and Stone ^[14]), (2.12) yields (2.11), the proof is completed.

Proof of Theorem 1.1. Theorem 1.1 can be obtained by combining Lemma 2.3 with Lemma 2.4 directly.

Proof of Corollary 1.2. By the strictly stationary and $0 < \sigma^2 = :EY_1^2+\sum_{k = 1}^{\infty}EY_1Y_{1+k} < \infty$, it is easy to see

Then Corollary 1.2 follows from Theorem 1.1.

3.   Conclusions

In this paper, using the Rosenthal type maximal inequality and Berry-Esseen bounds, the law of the iterated logarithm for a class of random variables is established, this extends the results of Cai and Wu ^[12] from NA case to general case, because that END and NSD random variables are much weaker than independent random variables and NA random variables thanking to Shen ^[7] and Shen et al. ^[9] for details.

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11771178) and Science and Technology Program of Jilin Educational Department during the "13th Five-Year'' Plan Period (Grant No. JJKH20200951KJ).

Conflict of interest

The authors declare no conflict of interest in this paper.