Abstract
Under suitable conditions, the almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. The simulation of the almost sure convergence for the maximum is firstly performed, which helps to visually understand the theorems by applying to two new examples.
1. Introduction
The classical extreme value theory studies extreme phenomena in nature and human society, such as floods, hurricane, stock market crash, megaseism, and so on [1–3]. Almost sure (a.s.) convergence is a nice behavior of the various ways of convergence [4, 5]. The almost sure central limit theorem (ASCLT) on partial sums is initiated by Brosamler and Schatte for independent identically distributed (i.i.d.) random variables [6, 7]. Let be i.i.d. random variables with and . Under some regularity conditions, we havefor any x, where , , and denote the indicator function, the natural logarithm, and the standard normal distribution function. Later, Lacey and Philipp prove that equation (1) holds when is a bound Lipschitz function [8]. The almost sure central limit theorem on maximum of i.i.d random variables is firstly discovered by Fahrnar and Stadtmüller and Cheng et al., respectively [5, 9]. Berkes et al. consider the general nonlinear functionals case [10]. Csáki and Gondigdanzan investigate ASCLT for the maximum of a stationary Gaussian sequences [11]. Chen and Lin extend to the nonstationary Gaussian sequences [12]. Chen and Peng provide an ASCLT for the maxima of multivariate stationary Gaussian sequences under some mild conditions [13, 14].
In recent years, the almost sure convergence of the joint distribution is explored. The almost sure limit theorems related to the maxima of the complete and incomplete samples for stationary sequences are considered [15]. Peng et al. extend the result to the maxima and minima of the complete and incomplete samples for weakly dependent stationary Gaussian sequences [16]. The ASCLT for the maxima and sums of i.i.d. random variables is investigated by Zang et al. [17]. Tan and Wang consider ASCLT for the maxima and sums of standardized stationary Gaussian sequences under some conditions [18]. The almost sure convergence of the maxima and minima for stationary sequences holds under some dependence conditions [19]. The behavior of almost sure convergence of the maxima and the minima for a strongly dependent stationary Gaussian sequence is extended to multivariate vectors [20, 21].
In this paper, we study ASCLT of the multivariate general standard normal sequences under some suitable conditions. Throughout this paper, is a standardized normal sequence of -dimensional random vector, i.e., each component of the random vector has a zero mean and a unit standard deviation. Setand and are two real vectors. implies for all , and stands for . Finally, write and . In this paper, the following results are obtained (proofs are given in Section 2).
2. Results
2.1. Theorems
Theorem 1. Let be a standardized normal sequence of -dimensional random vector, satisfying , for andas and
Let the levels be such that is bounded and for some . If for some and , then
In particular, let for , and then
Theorem 2. Let be a standardized normal -dimensional random vector sequence satisfying(a)(b)there exists , such thatwhere .
If then
Let , where are real numbers for , and then
We shall be concerned with the maxima for a normal sequence given by where is a standardized normal sequence of -dimensional vector with each component having a zero mean and unit standard deviation, and are added deterministic components. Define , and we shall assume that the constant is such that
Hence, the mean and the deviation of each component of are and 1, and
Let be chosen such that andwhere is a constant.
Theorem 3. Let be a normal sequence of -dimensional vector defined above with satisfying the conditions of Theorem 1. Then,where with and with and .
2.2. Proofs of the Main Results
We need the following lemmas for the proofs of the main results.
Lemma 1. Let and be standardized normal sequences of -dimensional random vector with , and , . Let , and be real vectors. Then,for constants .
Proof. It follows from Theorem 4.2.1 in [22].
Lemma 2. Let be a standardized normal sequence of -dimensional random vector satisfying equations (3) and (4), and be bounded for , then
Proof. By equation (3), we have . Define for . Split into two parts, thenNotice that is bounded for , which implies thatfor some constant , , and thenfor .
Now, consider . Let and . Since is bounded as such that for some and , and thenAswe haveand is bounded. Then,
Lemma 3. Let be a standardized normal -dimensional random vector sequence satisfying the conditions (a) and (b) of Theorem 2, and then equations (15) and (16) hold.
Proof. The proof of Lemma 3 is similar to that of Lemma 2.
Lemma 4. Let be a standardized normal sequence of -dimensional random vector satisfying Lemma 2 (Lemma 3), then
Proof. By Lemma 1,Both and are of order , but we will only prove . In fact,We have
Remark 1. Under the conditions of Theorem 2, Lemma 4 can be proved by Lemma 1 and Lemma 3.
Lemma 5. Let be a standardized normal sequence of -dimensional random vector satisfying Lemma 2 (Lemma 3), then where denotes the mean.
Proof. By Lemma 1 and Lemma 2, we haveMeanwhile, from which the lemma follows.
Remark 2. Under the conditions of Theorem 2, Lemma 5 also can be proved using Lemma 1 and Lemma 3.
Lemma 6. Suppose that is a standardized normal sequence of -dimensional random vector satisfying equation (3) of Theorem 1, andAs for and all , thenEspecially, let with for all ; then,
Proof. By Lemma 1, equation (32) can be obtained under the condition of equation (31), and the process of proof is similar to that of Lemma 2.
Lemma 7. Let be a standardized normal sequence of -dimensional random vector satisfying (a) of Theorem 3.
(c) There exists , as where .
If then
Let with for all , then
Proof. By Lemma 1, equation (36) can be obtained using equations (34) and (35), the method of proof is similar to that of Lemma 6
Lemma 8. Let be a sequence of bounded random variables. Ifthen
Proof. See Lemma 3.1 in [11].
Proof of Theorem. 1. Let , andBy Lemma 4 and Lemma 5,By Lemma 6 and Lemma 8, the proof of the theorem is completed.
Proof of Theorem. 2. Under the conditions of Theorem 2, using Lemma 4 and Lemma 5, we haveBy Lemma 7 and Lemma 8, Theorem 2 is obtained.
Proof of Theorem. 3. Write where , .By Theorem 1, it is needed to check that is bounded with . Notice that as , and , , andfor large . Hence, is bounded and satisfies the conditions of Theorem 1. As , Theorem 3 can be proved.
2.3. Simulation
The almost sure central limit theorems for the maximum of the multivariate normal sequences are complete. Simulating the almost sure convergence for the maxima, which visually describe the theorems, is not performed in previous papers [12, 13, 15, 16, 23–26]. To construct a suitable standardized nonstationary normal sequence is the key to the simulation.
Let be an independent random sequence from the standardized normal distribution . Set , then and , that is, , . As the correlation coefficient of and , , be a standardized nonstationary normal random variable sequence. For , the sequence satisfies equations (3) and (4). When are one-dimensional variables and , equation (6) of Theorem 2 holds.
Theorem 2 is simulated based on the standardized nonstationary normal random variable sequence by Matlab. denotes the number of random variable sequence and . , , especially , and . We note that is greater than 1 and converges to 1, but the rate of convergence is slow. The value and the number of the random variables are displayed in Table 1. The fitting effect gets better with the number of random variable increasing. The detailed simulation is shown in Figure 1.
Let , then , and , , . is a two-dimensional random vector.
is a standardized nonstationary normal sequence of two-dimensional random vector satisfying equations (3) and (4), and then Theorem 2 holds as is a 2-dimensional random vector. The detailed simulation is shown in Figure 2.
3. Conclusions
The extreme value theory deals with extreme phenomena which are less likely to occur, but more harmful [1–3]. Almost sure limit theorems, nice behavior of convergence, for the maxima of normal vector sequences are put forward and proved. Figures 1 and 2 display good simulation for two examples and show the rates of the convergence visually, which promote the intuitive understanding about the almost sure central limit theorems for the maximum of nonstationary normal sequences of random vectors.
Data Availability
All data generated or analyzed during this study are included within this manuscript. If necessary, other relevant data can be made available from the corresponding author via mail upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61374183 and 51535005), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-I-0418K01 and MCMS- I-0418Y01), the Fundamental Research Funds for the Central Universities (NC2018001, NP2019301, and NJ2019002), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Higher Education Institution Key Research Project Plan of Henan Province, China (20B110005).