On Chung-Teicher Type Strong Law of Large Numbers for 𝜌^∗-Mixing Random Variables

Abstract

In this paper the classical strong laws of large number of Kolmogorov, Chung, and Teicher for independent random variables were generalized on the case of 𝜌^∗-mixing sequence. The main result was applied to obtain a Marcinkiewicz SLLN.

1. Introduction

Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables defined on the probability space (Ω,ℱ,𝑃) with value in a real space ℝ and let 𝑆_𝑛=∑^𝑛_𝑖=1𝑋_𝑖. We say that the sequence {𝑋_𝑛,𝑛≥1} satisfies the strong law of large numbers (SLLN) if there exists some increasing sequence {𝑏_𝑛,𝑛≥1} and some sequence {𝑎_𝑛,𝑛≥1} such that∑^𝑛_𝑖=1𝑋_𝑖−𝑎_𝑖𝑏_𝑛⟶0a.s.as𝑛⟶∞. (1.1)In this paper, we consider the strong law of large numbers for sequences of dependent random variables which are said to be 𝜌^∗-mixing. To introduce the concept of 𝜌^∗-mixing sequence, we need the maximal correlation coefficient defined as follows:𝜌^∗(𝑘)=sup_𝑆,𝑇sup_{𝑋∈𝐿2(ℱ𝑆),𝑌∈𝐿2(ℱ𝑇)}cov(𝑋,𝑌)√Var𝑋⋅Var𝑌, (1.2)where 𝑆, 𝑇 are the finite subsets of positive integers such that dist (𝑆,𝑇)=inf_{𝑥∈𝑆,𝑦∈𝑇}|𝑥−𝑦|≥𝑘 and ℱ_𝑊 is the 𝜎-field generated by the random variables {𝑋_𝑖,𝑖∈𝑊⊂ℕ}.

Definition 1.1. A sequence of random variables {𝑋_𝑛,𝑛≥1} is said to be a 𝜌^∗-mixing sequence if lim_𝑛→∞𝜌^∗(𝑛)<1. (1.3)

𝜌^∗-mixing random variables were investigated by many authors. Various moment inequalities for sums and maximum of partial sums can be found in papers by Bradley [1], Bryc and Smoleński [2], Peligrad [3], Peligrad and Gut [4], and Utev and Peligrad [5]. These inequalities are used in many papers concerning the problems of invariance principle (Utev and Peligrad [5]), CLT (Peligrad [3]), or complete convergence for some stochastically dominated sequence of 𝜌^∗-mixing random variables (Cai [6]), and for an array of rowwise 𝜌^∗-mixing random variables (Zhu [7]). They will be also important in our further consideration.

The aim of this paper is to give some sufficient conditions for SLLN for a sequence of 𝜌^∗-mixing random variables without assumptions of identical distribution and stochastical domination. The result presented in this paper is obtained by using the maximal type inequality and the following strong law of large numbers proved by Fazekas and Klesov [8].

Theorem 1.2 (Fazekas and Klesov [8]). Let {𝑏_𝑛,𝑛≥1} be a nondecreasing, unbounded sequence of positive numbers. Let {𝛼_𝑛,𝑛≥1} be nonnegative numbers. Let 𝑟 be a fixed positive number. Assume that for each 𝑛≥1: 𝐸max_{1≤𝑖≤𝑛}𝑆_𝑖^𝑟≤^𝑛_𝑖=1𝛼_𝑖. (1.4)If^∞_𝑖=1𝛼_𝑖𝑏^𝑟_𝑖<∞, (1.5)thenlim_𝑛→∞𝑆_𝑛𝑏_𝑛=0a.s. (1.6)

Using this theorem, we are going to show that classical Kolmogorov, Chung, and Teicher's strong law of large numbers for independent random variables {𝑋_𝑛,𝑛≥1} (Chung [9] and Teicher [10]) can be generalized to the case of 𝜌^∗-mixing sequences.

In our further consideration, we need the following result.

Lemma 1.3 (Utev and Peligrad [5]). Let {𝑋_𝑛,𝑛≥1} be a 𝜌^∗-mixing sequence with 𝐸𝑋_𝑛=0, 𝐸|𝑋_𝑛|^𝑞<∞, 𝑛≥1, 𝑞≥2. Then, there exists a positive constant 𝑐 such that 𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑋_𝑖^𝑞≤𝑐^𝑛_𝑖=1𝐸𝑋_𝑖^𝑞+^𝑛_𝑖=1𝐸𝑋²_𝑖^𝑞/2,∀𝑛≥1. (1.7)

Let 𝐶 denote a constant which is not necessary the same in its each appearance.

2. The Main Result

Let {𝜑_𝑛,𝑛≥1} be a sequence of nonnegative, even, continuous and nondecreasing on (0,∞) functions 𝜑_𝑛∶ℝ→ℝ⁺ with lim_𝑥→∞𝜑_𝑛(𝑥)=∞ and such that for all 𝑛≥1 and 1<𝑝≤2:(a)𝜑_𝑛(𝑥)/𝑥↘or(b)𝜑_𝑛(𝑥)/𝑥↗,𝜑_𝑛(𝑥)/𝑥^𝑝↘as𝑥⟶∞. (2.1)

Theorem 2.1. Let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜌^∗-mixing random variables and let {𝑏_𝑛,𝑛≥1} be an increasing sequence of positive real numbers. Let 1<𝑝≤2.
Assume that {𝜑_𝑛,𝑛≥1} satisfies (𝑎) in (2.1) with
(A)(𝐴)^∞_𝑖=2𝑏_𝑖^−𝑝𝐸𝜑^𝑝_𝑖𝑋_𝑖𝜑^𝑝_𝑖𝑏_𝑖+𝜑^𝑝_𝑖𝑋_𝑖^𝑖−1_𝑗=1𝑏^𝑝_𝑗𝐸𝜑^𝑝_𝑗𝑋_𝑗𝜑^𝑝_𝑗𝑏_𝑗+𝜑^𝑝_𝑗𝑋_𝑗<∞, (2.2)(B)(𝐵)^∞_𝑛=1𝑃𝑋_𝑛≥𝑎_𝑛<∞, (2.3)
for some sequence {𝑎_𝑛,𝑛≥1} of positive numbers such that (C)(𝐶)^∞_𝑛=1𝐸𝜑_𝑛^2𝑝min𝑋_𝑛,𝑎_𝑛𝜑_𝑛^2𝑝𝑏_𝑛+𝜑_𝑛^2𝑝𝑋_𝑛<∞, (2.4) or {𝜑_𝑛,𝑛≥1} satisfies (𝑏) in (2.1) with (A₁)𝐴₁^∞_𝑖=2𝑏_𝑖^−𝑝𝐸𝜑_𝑖𝑋_𝑖𝜑_𝑖𝑏_𝑖+𝜑_𝑖𝑋_𝑖^𝑖−1_𝑗=1𝑏^𝑝_𝑗𝐸𝜑_𝑗𝑋_𝑗𝜑_𝑗𝑏_𝑗+𝜑_𝑗𝑋_𝑗<∞, (2.5) and (𝐵) for some sequence {𝑎_𝑛,𝑛≥1} of positive numbers such that (C₁)𝐶₁^∞_𝑛=1𝐸𝜑²_𝑛min𝑋_𝑛,𝑎_𝑛𝜑²_𝑛𝑏_𝑛+𝜑²_𝑛𝑋_𝑛<∞. (2.6)
Then,𝑏_𝑛^𝑛−1_𝑖=1𝑋_𝑖−𝐸𝑋_𝑖𝐼𝑋_𝑖<𝑏_𝑖⟶0a.s.𝑛⟶∞. (2.7)

Proof. Let 𝑋^′_𝑖=𝑋_𝑖𝐼[|𝑋_𝑖|<𝑏_𝑖], 𝑋^∗_𝑖=𝑋^′_𝑖−𝐸𝑋^′_𝑖, 𝑆^′_𝑛=∑^𝑛_𝑖=1𝑋^∗_𝑖, and 𝑆_𝑛=∑^𝑛_𝑖=1𝑋_𝑖.
Then,^∞_𝑛=1𝑃𝑋^′_𝑛≠𝑋_𝑛=^∞_𝑛=1𝑃𝑋_𝑛≥𝑏_𝑛≤^∞_𝑛=1𝑃2𝜑_𝑛^2𝑟𝑋_𝑛≥𝜑_𝑛^2𝑟𝑏_𝑛+𝜑_𝑛^2𝑟𝑋_𝑛≤𝐶^∞_𝑛=1𝐸𝜑_𝑛^2𝑟𝑋_𝑛𝜑_𝑛^2𝑟𝑏_𝑛+𝜑_𝑛^2𝑟𝑋_𝑛⋅𝐼|𝑋_𝑛|≥𝑎_𝑛𝜑+𝐸^𝑛2𝑟𝑋_𝑛𝜑_𝑛^2𝑟𝑏_𝑛+𝜑_𝑛^2𝑟𝑋_𝑛⋅𝐼𝑋_𝑛<𝑎_𝑛≤𝐶^∞_𝑛=1𝑃𝑋_𝑛≥𝑎_𝑛+^∞_𝑛=1𝐸𝜑_𝑛^2𝑟min𝑋_𝑛,𝑎_𝑛𝜑_𝑛^2𝑟𝑏_𝑛+𝜑_𝑛^2𝑟𝑋_𝑛<∞ (2.8)for 𝑟=𝑝 in the case (a) or 𝑟=1 in the case (b). Hence, the sequences {𝑋_𝑛,𝑛≥1} and {𝑋^′_𝑛,𝑛≥1} are equivalent in Khinchin's sense.
Thus, we need only to show that𝑆^′_𝑛𝑏_𝑛⟶0a.s.𝑛⟶∞. (2.9)By Lemma 1.3, for 𝑞=4, we have𝐸max_{1≤𝑖≤𝑛}𝑆^′_𝑖⁴≤𝐶^𝑛_𝑖=1𝐸𝑋^∗_𝑖⁴+^𝑛_𝑖=1𝐸𝑋^∗_𝑖²²≤𝐶2^𝑛_𝑖=1𝐸𝑋^∗_𝑖⁴+2^𝑛_𝑖=2𝐸𝑋^∗_𝑖^2𝑖−1_𝑗=1𝐸𝑋^∗_𝑗²≤𝐶𝐸𝑋^′₁⁴+^𝑛_𝑖=2𝐸𝑋^′_𝑖⁴+𝐸𝑋^′_𝑖^2𝑖−1_𝑗=1𝐸𝑋^′_𝑗². (2.10)By Theorem 1.2 applied with𝛼₁=𝐸𝑋^′₁⁴,𝛼_𝑖=𝐸𝑋^′_𝑖⁴+𝐸𝑋^′_𝑖^2𝑖−1_𝑗=1𝐸𝑋^′_𝑗²for𝑖=2,3,…,𝑛, (2.11)we see that in order to show (2.9) it is enough to prove that^∞_𝑖=1𝛼_𝑖𝑏⁴_𝑖<∞, (2.12)which holds if^∞_𝑖=1𝐸𝑋^′_𝑖⁴𝑏⁴_𝑖<∞,^∞_𝑖=2𝐸𝑋^′_𝑖²𝑏⁴_𝑖^𝑖−1_𝑗=1𝐸𝑋^′_𝑗²<∞. (2.13)Put 𝐼∶=∑^∞_𝑖=1𝑃[|𝑋_𝑖|≥𝑎_𝑖]. Then, we have^∞_𝑖=1𝐸𝑋^′_𝑖⁴𝑏⁴_𝑖≤^∞_𝑖=1𝐸𝑋_𝑖𝐼𝑋_𝑖<min𝑎_𝑖,𝑏_𝑖⁴𝑏⁴_𝑖+𝐼≤^∞_𝑖=1𝐸𝑋_𝑖^2𝑝𝐼𝑋_𝑖<min𝑎_𝑖,𝑏_𝑖𝑏_𝑖^2𝑝+𝐼. (2.14)Moreover, we note that𝑎𝑏𝑎⋅𝐼[𝑎≤𝑏]≤2⋅𝑎+𝑏∀𝑎,𝑏>0. (2.15)Hence, in case (a) we get, by (B) and (C),^∞_𝑖=1𝐸𝑋^′_𝑖⁴𝑏⁴_𝑖≤^∞_𝑖=1𝐸𝜑_𝑖^2𝑝𝑋_𝑖𝜑_𝑖^2𝑝𝑏_𝑖𝐼𝑋_𝑖<min𝑎_𝑖,𝑏_𝑖+𝐼≤2^∞_𝑖=1𝐸𝜑_𝑖^2𝑝min𝑋_𝑖,𝑎_𝑖𝜑_𝑖^2𝑝𝑏_𝑖+𝜑_𝑖^2𝑝𝑋_𝑖+𝐼<∞, (2.16)while in case (b) we get, by (B) and (C₁),^∞_𝑖=1𝐸𝑋^′_𝑖⁴𝑏⁴_𝑖≤2^∞_𝑖=1𝐸𝜑²_𝑖min𝑋_𝑖,𝑎_𝑖𝜑²_𝑖𝑏_𝑖+𝜑²_𝑖𝑋_𝑖+𝐼<∞. (2.17)Using the fact that𝑏_𝑖⁻⁴𝐸𝑋^′_𝑖²𝐸𝑋^′_𝑗²≤𝑏_𝑖^−2𝑝𝐸𝑋^′_𝑖^𝑝𝐸𝑋^′_𝑗^𝑝∀𝑗<𝑖 (2.18)both in either case (a) ^∞_𝑖=2𝐸𝑋^′_𝑖²𝑏⁴_𝑖^𝑖−1_𝑗=1𝐸𝑋^′_𝑗²≤^∞_𝑖=2𝐸𝑋^′_𝑖^𝑝𝑏_𝑖^{2𝑝𝑖−1}_𝑗=1𝐸𝑋^′_𝑗^𝑝≤4^∞_𝑖=2𝑏_𝑖^−𝑝𝐸𝜑^𝑝_𝑖𝑋_𝑖𝜑^𝑝_𝑖𝑏_𝑖+𝜑^𝑝_𝑖𝑋_𝑖^𝑖−1_𝑗=1𝑏^𝑝_𝑗𝐸𝜑^𝑝_𝑗𝑋_𝑗𝜑^𝑝_𝑗𝑏_𝑗+𝜑^𝑝_𝑗𝑋_𝑗<∞, (2.19)or case (b) ^∞_𝑖=2𝐸𝑋^′_𝑖²𝑏⁴_𝑖^𝑖−1_𝑗=1𝐸𝑋^′_𝑗²≤^∞_𝑖=2𝐸𝑋^′_𝑖^𝑝𝑏_𝑖^{2𝑝𝑖−1}_𝑗=1𝐸𝑋^′_𝑗^𝑝≤4^∞_𝑖=2𝑏_𝑖^−𝑝𝐸𝜑_𝑖𝑋_𝑖𝜑_𝑖𝑏_𝑖+𝜑_𝑖𝑋_𝑖^𝑖−1_𝑗=1𝑏^𝑝_𝑗𝐸𝜑_𝑗𝑋_𝑗𝜑_𝑗𝑏_𝑗+𝜑_𝑗𝑋_𝑗∞. (2.20)Thus, we have established (2.9) and consequently (2.7).

Corollary 2.2. Let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜌^∗-mixing random variables satisfying the condition ^∞_𝑖=1𝐸𝑋_𝑖^𝑟𝑡𝑖^𝑟+𝑋_𝑖^𝑟𝑡<∞ (2.21)with 𝑟=𝑝 for 0<𝑡<1 and all 1<𝑝≤2, or 𝑟=2/𝑡 for 1≤𝑡<2.
Then,𝑛^{𝑛−1/𝑡}_𝑖=1𝑋_𝑖−𝐸𝑋_𝑖𝐼𝑋_𝑖<𝑖^1/𝑡⟶0a.s.as𝑛⟶∞, (2.22)

Proof. Let 𝑎_𝑛=𝑏_𝑛=𝑛^1/𝑡 for any 0<𝑡<2. Then, the assumption (B) of Theorem 2.1 is fulfilled.
Indeed we see (B)∶^∞_𝑛=1𝑃𝑋_𝑛≥𝑎_𝑛=^∞_𝑛=1𝑃𝑋_𝑛≥𝑛^1/𝑡≤𝐶^∞_𝑛=1𝑃2𝑋_𝑛^𝑟𝑡≥𝑛^𝑟+𝑋_𝑛^𝑟𝑡<𝐶⋅^∞_𝑛=1𝐸𝑋_𝑛^𝑟𝑡𝑛^𝑟+𝑋_𝑛^𝑟𝑡<∞ (2.23) by (2.21) with 𝑟=𝑝 for 0<𝑡<1 or 𝑟=2/𝑡 for 1≤𝑡<2.
Let now 0<𝑡<1. Then, for 𝜑_𝑛(𝑥)=𝑥^𝑡, 𝑥>0, 𝑛≥1, the conditions (A) and (C) with 1<𝑝≤2 are fulfilled: (A)∶^∞_𝑖=2𝑏_𝑖^−𝑝𝐸𝜑^𝑝_𝑖𝑋_𝑖𝜑^𝑝_𝑖𝑏_𝑖+𝜑^𝑝_𝑖𝑋_𝑖^𝑖−1_𝑗=1𝑏^𝑝_𝑗𝐸𝜑^𝑝_𝑗𝑋_𝑗𝜑^𝑝_𝑗𝑏_𝑗+𝜑^𝑝_𝑗𝑋_𝑗≤^∞_𝑖=1𝐸𝑋_𝑖^𝑝𝑡𝑖^𝑝+𝑋_𝑖^𝑝𝑡2<∞, (2.24) by (2.21) with 𝑟=𝑝. (C)∶^∞_𝑛=1𝐸𝜑_𝑛^2𝑝min𝑋_𝑛,𝑎_𝑛𝜑_𝑛^2𝑝𝑏_𝑛+𝜑_𝑛^2𝑝𝑋_𝑛=^∞_𝑛=1𝐸min𝑋_𝑛,𝑎_𝑛^2𝑝𝑡𝑛^2𝑝+𝑋_𝑛^2𝑝𝑡≤^∞_𝑛=1𝐸𝑋_𝑛^2𝑝𝑡𝑛^2𝑝+𝑋_𝑛^2𝑝𝑡𝐼𝑋_𝑛≥𝑛^1/𝑡+^∞_𝑛=1𝐸𝑋_𝑛^2𝑝𝑡𝑋_𝑛^𝑝𝑡𝑛^𝑝/𝑋_𝑛^𝑝𝑡⋅𝑛^𝑝+𝑋_𝑛^𝑝𝑡𝐼𝑋_𝑛<𝑛^1/𝑡≤𝐶^∞_𝑛=1𝑃𝑋_𝑛≥𝑛^1/𝑡+^∞_𝑛=1𝐸𝑋_𝑛^𝑝𝑡𝑛^𝑝+𝑋_𝑛^𝑝𝑡<∞ (2.25) by (2.23) and (2.21) with 𝑟=𝑝, 1<𝑝≤2.
Thus, by Theorem 2.1, we have𝑛^{𝑛−1/𝑡}_𝑖=1𝑋_𝑖−𝐸𝑋_𝑖𝐼𝑋_𝑖<𝑖^1/𝑡⟶0a.s.as𝑛⟶∞ (2.26)for 0<𝑡<1.
Now, we need to show that (2.26) also holds for 1≤𝑡<2.
Let 𝜑_𝑛(𝑥)=𝑥². Then, for 𝑝=2, by the similar calculations as in case 0<𝑡<1, we get A₁∶^∞_𝑖=2𝑏_𝑖^−𝑝𝐸𝜑_𝑖𝑋_𝑖𝜑_𝑖𝑏_𝑖+𝜑_𝑖𝑋_𝑖^𝑖−1_𝑗=1𝑏^𝑝_𝑗𝐸𝜑_𝑗𝑋_𝑗𝜑_𝑗𝑏_𝑗+𝜑_𝑗𝑋_𝑗≤^∞_𝑖=1𝐸𝑋_𝑖²𝑖^2/𝑡+𝑋_𝑖²²<∞ (2.27) by (2.21) with 𝑟=2/𝑡 and C₁∶^∞_𝑛=1𝐸𝜑²_𝑛min𝑋_𝑛,𝑎_𝑛𝜑²_𝑛𝑏_𝑛+𝜑²_𝑛𝑋_𝑛=^∞_𝑛=1𝐸min𝑋_𝑛,𝑎_𝑛⁴𝑛^4/𝑡+𝑋_𝑛⁴≤𝐶^∞_𝑛=1𝑃𝑋_𝑛≥𝑛^1/𝑡+^∞_𝑛=1𝐸𝑋_𝑛²𝑛^2/𝑡+𝑋_𝑛²<∞ (2.28) by (2.23) and (2.21) with 𝑟=2/𝑡.
Therefore, by Theorem 2.1, we get (2.26) for 1≤𝑡<2.
This completes the proof of Corollary 2.2.

Corollary 2.3. Let {𝑋,𝑋_𝑛,𝑛≥1} be a sequence of 𝜌^∗-mixing random variables with 𝐸|𝑋|^𝑡<∞, 0<𝑡<2. Let 𝑃𝑋_𝑖>𝑥≤𝐶𝑃|𝑋|>𝑥 (2.29)for all 𝑥>0, 𝑖≥1 and some positive constant 𝐶.
Moreover, when 1≤𝑡<2, let 𝐸𝑋=0. Then,𝑛^{𝑛−1/𝑡}_𝑖=1𝑋_𝑖⟶0a.s.as𝑛⟶∞. (2.30)

Proof. We first note that (2.29) implies𝐸𝑋_𝑖^𝑠𝐼𝑋_𝑖<𝑎≤𝐶𝐸|𝑋|^𝑠𝐼|𝑋|<𝑎+𝑎^𝑠𝑃|𝑋|≥𝑎 (2.31)for any 𝑎>0 and 𝑠>0.
Put now 𝑎_𝑛=𝑏_𝑛=𝑛^1/𝑡 for any 0<𝑡<2.
It is easy to see that 𝐸|𝑋|^𝑡<∞, (2.29) and (2.31) with 𝑠=𝑟𝑡 imply convergence of the series:^∞_𝑛=1𝐸𝑋_𝑛^𝑟𝑡𝑛^𝑟+𝑋_𝑛^𝑟𝑡 (2.32)for 𝑟=𝑝, (1<𝑝≤2) in the case 0<𝑡<1 and 𝑟=2/𝑡 in the case 1≤𝑡<2.
We have^∞_𝑛=1𝐸𝑋_𝑛^𝑟𝑡𝑛^𝑟+𝑋_𝑛^𝑟𝑡=^∞_𝑛=1𝐸𝑋_𝑛^𝑟𝑡𝑛^𝑟+𝑋_𝑛^𝑟𝑡𝐼𝑋_𝑛<𝑛^1/𝑡+^∞_𝑛=1𝐸𝑋_𝑛^𝑟𝑡𝑛^𝑟+𝑋_𝑛^𝑟𝑡𝐼𝑋_𝑛≥𝑛^1/𝑡≤^∞_𝑛=1𝐸𝑋_𝑛^𝑟𝑡𝑛^𝑟𝐼𝑋_𝑛<𝑛^1/𝑡+^∞_𝑛=1𝑃𝑋_𝑛≥𝑛^1/𝑡≤𝐶^∞_𝑛=1𝐸|𝑋|^𝑟𝑡𝑛^𝑟𝐼|𝑋|<𝑛^1/𝑡+^∞_𝑛=1𝑃|𝑋|≥𝑛^1/𝑡≤𝐶^∞_𝑛=1𝐸|𝑋|^{𝑡+𝑡(𝑟−1)}𝐼|𝑋|<𝑛^1/𝑡𝑛^{1+1/𝑡⋅𝑡(𝑟−1)}+𝐸|𝑋|^𝑡<∞. (2.33)Because of Corollary 2.2, this proves that (2.26) holds for any 0<𝑡<2.
To complete the proof we should show that𝑛^{−1/𝑡𝑛}_𝑖=1𝐸𝑋_𝑖𝐼𝑋_𝑖<𝑖^1/𝑡⟶0as𝑛⟶∞ (2.34)for any 0<𝑡<2.
For 0<𝑡<1 and 𝐼₁∶=𝑛^−1/𝑡∑^𝑛_𝑖=1𝑖^1/𝑡𝑃[|𝑋|≥𝑖^1/𝑡], we have𝑛^{−1/𝑡𝑛}_𝑖=1𝐸𝑋_𝑖𝐼𝑋_𝑖<𝑖^1/𝑡≤𝑛^{𝑛−1/𝑡}_𝑖=1𝐸𝑋_𝑖𝐼𝑋_𝑖<𝑖^1/𝑡≤𝐶𝑛^{𝑛−1/𝑡}_𝑖=1𝐸|𝑋|𝐼|𝑋|<𝑖^1/𝑡+𝐼₁≤𝐶𝑛^{𝑛−1/𝑡}_𝑖=1𝐸|𝑋|𝐼|𝑋|<𝑛^1/𝑡+𝐼₁≤𝐶𝑛^1−1/𝑡𝐸|𝑋|𝐼|𝑋|<𝑛^1/𝑡+𝐼₁=𝐶𝑛^{𝑛1−1/𝑡}_𝑗=1𝐸|𝑋|𝐼𝑗−1≤|𝑋|^𝑡<𝑗+𝐼₁. (2.35)But^∞_𝑛=1𝑃|𝑋|≥𝑛^1/𝑡≤𝐶𝐸|𝑋|^𝑡<∞, (2.36)so by Kronecker's lemma we get𝐼₁∶=𝑛^{𝑛−1/𝑡}_𝑖=1𝑖^1/𝑡𝑃|𝑋|≥𝑖^1/𝑡⟶0as𝑛⟶∞. (2.37)Moreover, we note^∞_𝑗=1𝑗^1−1/𝑡𝐸|𝑋|𝐼𝑗−1≤|𝑋|^𝑡<𝑗≤^∞_𝑗=1𝐸|𝑋|𝐼𝑗−1≤|𝑋|^𝑡<𝑗=𝐸|𝑋|^𝑡<∞, (2.38)and by Kronecker's lemma𝑛^{𝑛1−1/𝑡}_𝑗=1𝐸|𝑋|𝐼𝑗−1≤|𝑋|^𝑡<𝑗⟶0as𝑛⟶∞, (2.39)which together with (2.35) and (2.37) gives (2.34) for 0<𝑡<1.
Let 1≤𝑡<2. First, we will show thatlim_𝑛→∞𝑃^𝑛_𝑖=1𝑋_𝑖>𝜀𝑛^1/𝑡=0. (2.40)To achieve this, we put 𝑌_𝑖=𝑋_𝑖𝐼[|𝑋_𝑖|<𝑛^1/𝑡] for 1≤𝑖≤𝑛.
Because of 𝐸𝑋_𝑖=0 and 𝐸|𝑋|^𝑡<∞, we have𝑛^{−1/𝑡𝑛}_𝑖=1𝐸𝑌_𝑖≤𝑛^{𝑛−1/𝑡}_𝑖=1𝐸𝑋_𝑖𝐼𝑋_𝑖≥𝑛^1/𝑡≤𝐶𝐸|𝑋|^𝑡𝐼|𝑋|≥𝑛^1/𝑡⟶0as𝑛⟶∞. (2.41)Therefore, for large enough 𝑛, we obtain𝑃^𝑛_𝑖=1𝑋_𝑖>𝜀𝑛^1/𝑡≤^𝑛_𝑖=1𝑃𝑋_𝑖≥𝑛^1/𝑡+𝑃^𝑛_𝑖=1(𝑌_𝑖−𝐸𝑌_𝑖)>𝜀𝑛^1/𝑡. (2.42)Hence, we need only to show that^𝑛_𝑖=1𝑃𝑋_𝑖≥𝑛^1/𝑡⟶0as𝑛⟶∞,𝑃^𝑛_𝑖=1𝑌_𝑖−𝐸𝑌_𝑖>𝜀𝑛^1/𝑡⟶0as𝑛⟶∞. (2.43)(2.44)By 𝐸|𝑋|^𝑡<∞, we get^∞_𝑛=11𝑛^𝑛_𝑖=1𝑃𝑋_𝑖≥𝑛^1/𝑡≤𝐶^∞_𝑛=1𝑃|𝑋|≥𝑛^1/𝑡≤𝐶𝐸|𝑋|^𝑡<∞, (2.45)which implies (2.43).
By Lemma 1.3 and (2.23) with 𝑠=2,^∞_𝑛=11𝑛𝑃^𝑛_𝑖=1𝑌_𝑖−𝐸𝑌_𝑖>𝜀𝑛^1/𝑡≤𝐶^∞_𝑛=1𝑛^{𝑛−(1+2/𝑡)}_𝑖=1𝐸𝑌²_𝑖≤𝐶^∞_𝑛=1𝑛^{𝑛−(1+2/𝑡)}_𝑖=1𝐸𝑋²𝐼|𝑋|<𝑛^1/𝑡+^∞_𝑛=1𝑛^{𝑛−(1+2/𝑡)}_𝑖=1𝑛^2/𝑡𝑃|𝑋|≥𝑛^1/𝑡≤𝐶^∞_𝑛=1𝑛^{𝑛−2/𝑡}_𝑗=1𝐸𝑋²𝐼𝑗−1≤|𝑋|^𝑡<𝑗+^∞_𝑛=1𝑃|𝑋|≥𝑛^1/𝑡≤𝐶^∞_𝑘=1𝐸𝑋²𝐼𝑘−1≤|𝑋|^𝑡<𝑘^∞_𝑛=𝑘𝑛^−2/𝑡+𝐸|𝑋|^𝑡≤𝐶^∞_𝑘=1𝑘^−2/𝑡+1𝐸𝑋²𝐼𝑘−1≤|𝑋|^𝑡<𝑘+𝐸|𝑋|^𝑡≤𝐶^∞_𝑘=1𝑘𝑃𝑘−1≤|𝑋|^𝑡<𝑘+𝐸|𝑋|^𝑡<𝐶𝐸|𝑋|^𝑡<∞ (2.46)which gives (2.44).
Thus, we have established that (2.40) holds true. Equations (2.40) and (2.26) imply (2.34) which completes the proof of Corollary 2.3.

Corollary 2.3 gives the Marcinkiewicz SLLN for 𝜌^∗-mixing random variables {𝑋_𝑛,𝑛≥1} stochastically dominated by a random variable 𝑋 for 0<𝑡<2. The identical result was obtained by Cai [6] as a consequence of complete convergence theorem. Both of them, for 0<𝑡<1, are a special case of more general result of Fazekas and Tómács [11] obtained for stochastically dominated random variables without assuming any kind of dependence.

Acknowledgment

The author thanks the referees for their comments and suggestions which allowed to improve this paper.