- Open Access
Exponential inequalities for self-normalized martingales
© Chen et al.; licensee Springer 2014
- Received: 27 April 2014
- Accepted: 19 July 2014
- Published: 18 August 2014
In this paper, we establish several exponential inequalities for martingales and self-normalized martingales, which improve some known results.
- exponential inequalities
- self-normalized martingale difference sequence
- random variables
- tail probabilities
A prototypical example of self-normalized random variables is Student’s t-statistic which replaces the population standard deviation σ in the standardized sample mean by the sample standard deviation. Generally, a self-normalized process is of the form , in which is a random variable that estimates some dispersion measure of . If is normalized by nonrandom instead, the moment conditions are needed. However, for self-normalized process, the moment conditions can be deleted, for example, Shao  obtained the large deviation of self-normalized sums of i.i.d. random variables without moment conditions. In addition, there has been increasing interest in limit theorems and moment bounds for self-normalized sums of i.i.d. zero-mean random variables . Bentkus and Götze  gave a Berry-Esseen bound for Student’s t-statistic. Giné et al.  proved that the t-statistic has a limiting standard normal distribution if and only if is in the domain of attraction of a normal law. We refer to De la Peña et al.  for the comprehensive review of the state of the art of the theory and it applications in statistical inference.
where . In , Bercu and Touati established the following results without any assumptions on .
Theorem 1.1 [, Theorem 2.1]
Theorem 1.2 [, Theorem 2.2]
then are called satisfying canonical assumption. For such a pair , De la Pen̂a and Pang  proved the following exponential bounds (in their inequalities, there are some misprints, so we state a correction as follows).
Theorem 1.3 [, Theorem 2.1]
Here we want to mention the works in Bercu , which provided some other different inequality forms.
The purpose of this paper is to establish several exponential inequalities motivated by the above works. In Section 2, we shall propose exponential inequalities for martingales and self-normalized martingales, and, in the last section, the deviation bound of the least-square estimate of the unknown parameter in a linear regressive model is established.
2.1 Exponential inequalities for martingales
In this subsection we will give some exponential inequalities involving and .
the inequality (3) is more precise than the inequality (1). Therefore, Theorem 2.1 improves the works of Bercu and Touati [, Theorem 2.1].
For self-normalized martingales, we can obtain the following inequality which improves the well-known Theorem 1.1.
Hence, Theorem 2.2 improves Theorem 1.2.
Therefore, we can obtain the following theorem.
where , and .
Remark 2.3 By Fatou’s lemma, it is easy to see that the above results still hold for any stopping time T with respect the filtration with a.s.
The following result may be of independent interest.
From Theorem 2.4, we have the following.
2.2 Proofs of main results
We start with the following basic lemma.
So we get , which means that is concave function. Because and , we can prove , i.e. . Finally we prove Lemma 2.1 by (9). □
In order to prove our main results, we firstly introduce Lemma 2.2.
Then, for all , is a positive supermartingale with .
As a result, for all , is a positive supermartingale, i.e. for all , , which implies that . □
Next, we start to prove Theorem 2.1 and Theorem 2.2 inspired by the original article of Bercu and Touati .
We also have the same upper bound for , immediately leading to the result of (3). □
We also find the same upper bound for , which completes the proof of Theorem 2.2. □
Since for , then we have , which, together with (18), yields the desired result. □
Bercu and Touati  obtained the following result.
Corollary 3.1 [, Corollary 5.1]
Now, we give the following theorem.
Remark 3.1 Obviously, the upper bound in (22) is better than the bound (21).
then, from the above discussions, the desired results can be obtained. □
This work is supported by IRTSTHN (14IRTSTHN023), NSFC (No. 11001077), NCET (NCET-11-0945), and Plan For Scientific Innovation Talent of Henan Province (124100510014).
- Shao QM: Self-normalized large deviations. Ann. Probab. 1997, 25: 285–328.MathSciNetView ArticleMATHGoogle Scholar
- Bentkus V, Götze F: The Berry-Esseen bound for Student’s statistic. Ann. Probab. 1996, 24: 491–503.MathSciNetView ArticleMATHGoogle Scholar
- Giné M, Götze F, Mason DM: When is the Student t -statistic asymptotically standard normal? Ann. Probab. 1997, 25: 1514–1531.MathSciNetView ArticleMATHGoogle Scholar
- De la Peña VH, Lai TL, Shao QM: Self-Normalized Processes: Limit Theory and Statistical Applications. Springer, Berlin; 2009.View ArticleMATHGoogle Scholar
- Bercu B, Touati A: Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 2008, 18: 1848–1869. 10.1214/07-AAP506MathSciNetView ArticleMATHGoogle Scholar
- De la Peña VH, Pang GD: Exponential inequalities for self-normalized processes with applications. Electron. Commun. Probab. 2009, 14: 372–381.MathSciNetView ArticleMATHGoogle Scholar
- Bercu B: An exponential inequality for autoregressive processes in adaptive tracking. J. Syst. Sci. Complex. 2007, 20: 243–250. 10.1007/s11424-007-9021-6MathSciNetView ArticleMATHGoogle Scholar
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