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A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums

Abstract

Let be a sequence of independent and identically distributed (i.i.d.) random variables and denote , . In this paper, we investigate the almost sure central limit theorem in the joint version for the maxima and sums. If for some numerical sequences , we have for a nondegenerate distribution , and is a bounded Lipschitz 1 function, then almost surely, where stands for the standard normal distribution function, ,and , .

1. Introduction and Main Results

Let be a sequence of independent and identically distributed (i.i.d.) random variables and , for . If , the classical almost sure central limit theorem (ASCLT) has the simplest form as follows:

(1.1)

almost surely for all , here and in the sequel, is the indicator function of the event , and stands for the standard normal distribution function. This result was first proved independently by Brosamler [1] and Schatte [2] under a stronger moment condition, since then, this type of almost sure version which mainly dealt with logarithmic average limit theorems has been extended in various directions. Fahrner and Stadtmüller [3] and Cheng et al. [4] extended this almost sure convergence for partial sums to the case of maxima of i.i.d. random variables. Under some natural conditions, they proved that

(1.2)

almost surely for all , where and satisfy

(1.3)

for any continuity point of .

For Gaussian sequences, Csáki and Gonchigdanzan [5] investigated the validity of (1.2) maxima of stationary Gaussian sequences under some mild condition. Furthermore, Chen and Lin [6] extended it to nonstationary Gaussian sequences. As for some other dependent random variables, Peligrad and Shao [7] and Dudziński [8] derived some corresponding results about ASCLT. The almost sure central limit theorem in the joint version for log average in the case of independent and identically distributed random variables is obtained by Peng et al. [9]; a joint version of almost sure limit theorem for log average of maxima and partial sums in the case of stationary Gaussian random variables is derived by Dudziński [10].

All the above results are related to the almost sure logarithmic version; in this paper; inspired by the results of Berkes and Csáki [11], we further study ASCLT in the joint version for the maxima and partial sums with another weighted sequence (). Now, we state our main result as follows.

Theorem 1.1.

Let be a sequence of independent and identically distributed (i.i.d.) random variables with non-degenerate common distribution function , satisfying and . If for some numerical sequences , one has

(1.4)

for a non-degenerate distribution , and is a bounded Lipschitz 1 function, then

(1.5)

almost surely, where stands for the standard normal distribution function, and , .

Remark 1.2.

Since a set of bounded Lipschitz 1 functions is tight in a set of bounded continuous functions, Theorem 1.1 is true for all bounded continuous functions .

Remark 1.3.

Under the conditions of Theorem 1.1, it can be seen that the result for indicator functions by routine approximation arguments is similar, for example, to those in Lacey and Philipp [12], that is,

(1.6)

almost surely.

Example 1.4.

The ASCLT has already received applications in many fields, including condensed matter physics, statistical mechanics, ergodic theory and dynamical systems, and control and information and quanitle estimation. As an example, we assume that is a sequence of independent and identically distributed (i.i.d.) random variables with standard normal distribution function , and in (1.4), we choose

(1.7)
(1.8)

which imply that (see Leadbetter et al. [13, Theorem ])

(1.9)

where is one of the extreme value distributions, that is,

(1.10)

Then, we can derive a corresponding result in Theorem 1.1.

2. Proof of Our Main Result

In this section, denote and , for , unless it is specially mentioned. Here and stand for and , respectively. is the standard normal distribution function.

Proof of Theorem 1.1.

Firstly, by Theorem 1.1. in [14] and our assumptions, we have

(2.1)

for . Then, in view of the dominated convergence theorem, we have

(2.2)

Hence, to complete the proof, it is sufficient to show that

(2.3)

almost surely. Let

(2.4)

For , it follows that

(2.5)

For , by the independence of , we have

(2.6)

Now, we are in a position to estimate . We use the fact that is bounded and Lipschitzian, then it follows that

(2.7)

Via Cauchy-Schwarz inequality, we have

(2.8)

Thus, using (2.6), (2.7), and (2.8), it follows that

(2.9)

for . Then, we have

(2.10)

It is obvious that

(2.11)

In view of the definition of numerical sequence and by L'Hospital rule and fixed , we have

(2.12)

Then

(2.13)

for such that From (2.11), (2.13), and the Markov inequality, we derive

(2.14)

for the above and . We can choose subsequence , where such that Then, by Borel-Cantelli lemma, we derive

(2.15)

almost surely. For we have

(2.16)

almost surely. Since , the convergence of the subsequence implies that the whole sequence converges almost surely. Hence the proof of (1.5) is completed for . Via the same arguments, we can obtain (1.5) for .

References

  1. Brosamler GA: An almost everywhere central limit theorem. Mathematical Proceedings of the Cambridge Philosophical Society 1988, 104(3):561–574. 10.1017/S0305004100065750

    MathSciNet  Article  MATH  Google Scholar 

  2. Schatte P: On strong versions of the central limit theorem. Mathematische Nachrichten 1988, 137: 249–256. 10.1002/mana.19881370117

    MathSciNet  Article  MATH  Google Scholar 

  3. Fahrner I, Stadtmüller U: On almost sure max-limit theorems. Statistics & Probability Letters 1998, 37(3):229–236. 10.1016/S0167-7152(97)00121-1

    MathSciNet  Article  MATH  Google Scholar 

  4. Cheng S, Peng L, Qi Y: Almost sure convergence in extreme value theory. Mathematische Nachrichten 1998, 190: 43–50. 10.1002/mana.19981900104

    MathSciNet  Article  MATH  Google Scholar 

  5. Csáki E, Gonchigdanzan K: Almost sure limit theorems for the maximum of stationary Gaussian sequences. Statistics & Probability Letters 2002, 58(2):195–203. 10.1016/S0167-7152(02)00128-1

    MathSciNet  Article  MATH  Google Scholar 

  6. Chen S, Lin Z: Almost sure max-limits for nonstationary Gaussian sequence. Statistics & Probability Letters 2006, 76(11):1175–1184. 10.1016/j.spl.2005.12.018

    MathSciNet  Article  MATH  Google Scholar 

  7. Peligrad M, Shao QM: A note on the almost sure central limit theorem for weakly dependent random variables. Statistics & Probability Letters 1995, 22(2):131–136. 10.1016/0167-7152(94)00059-H

    MathSciNet  Article  MATH  Google Scholar 

  8. Dudziński M: A note on the almost sure central limit theorem for some dependent random variables. Statistics & Probability Letters 2003, 61(1):31–40. 10.1016/S0167-7152(02)00291-2

    MathSciNet  Article  MATH  Google Scholar 

  9. Peng Z, Wang L, Nadarajah S: Almost sure central limit theorem for partial sums and maxima. Mathematische Nachrichten 2009, 282(4):632–636. 10.1002/mana.200610760

    MathSciNet  Article  MATH  Google Scholar 

  10. Dudziński M: The almost sure central limit theorems in the joint version for the maxima and sums of certain stationary Gaussian sequences. Statistics & Probability Letters 2008, 78(4):347–357. 10.1016/j.spl.2007.07.007

    MathSciNet  Article  MATH  Google Scholar 

  11. Berkes I, Csáki E: A universal result in almost sure central limit theory. Stochastic Processes and Their Applications 2001, 94(1):105–134. 10.1016/S0304-4149(01)00078-3

    MathSciNet  Article  MATH  Google Scholar 

  12. Lacey MT, Philipp W: A note on the almost sure central limit theorem. Statistics & Probability Letters 1990, 9(3):201–205. 10.1016/0167-7152(90)90056-D

    MathSciNet  Article  MATH  Google Scholar 

  13. Leadbetter MR, Lindgren G, Rootzén H: Extremes and Related Properties of Random Sequences and Processes, Springer Series in Statistics. Springer, New York, NY, USA; 1983:xii+336.

    Book  MATH  Google Scholar 

  14. Hsing T: A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables. The Annals of Probability 1995, 23(2):938–947. 10.1214/aop/1176988296

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

The authors thank two anonymous referees for their useful comments. They would like to thank professor Zhengyan Lin of Zhejiang University in China for his help. The work has been supported by the Young Excellent Talent Foundation of Huaiyin Normal University.

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Correspondence to Qing-pei Zang.

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Zang, Qp., Wang, Zx. & Fu, Ka. A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums. J Inequal Appl 2010, 234964 (2010). https://doi.org/10.1155/2010/234964

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Keywords

  • Central Limit Theorem
  • Gaussian Random Variable
  • Dominate Convergence Theorem
  • Numerical Sequence
  • Dependent Random Variable