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Exponential inequalities for self-normalized martingales
Journal of Inequalities and Applications volume 2014, Article number: 289 (2014)
In this paper, we establish several exponential inequalities for martingales and self-normalized martingales, which improve some known results.
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.
In this paper, we concentrate on the exponential inequalities of the self-normalized martingale. Let be a locally square integrable real martingale adapted to a filtration with . The predictable quadratic variation and the total quadratic variation of are, respectively, given by
where . In , Bercu and Touati established the following results without any assumptions on .
Theorem 1.1 [, Theorem 2.1]
Let be a locally square integrable martingale. Then, for all ,
Theorem 1.2 [, Theorem 2.2]
Let be a locally square integrable martingale. Then, for all , and ,
Moreover, we also have
It is necessary to point out that, to calculate the above exponential bounds, the following canonical assumption is important. For a pair of random variables with , if the following inequality holds:
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]
Let be a pair of random variables with in the probability space satisfying the canonical assumption. Suppose for some . Then, for any and for such that ,
In particular, if ,
and if ,
Moreover, if B satisfies , the upper bound becomes
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 Exponential inequalities
2.1 Exponential inequalities for martingales
In this subsection we will give some exponential inequalities involving and .
Theorem 2.1 Let be a locally square integrable martingale. Then, for all ,
Remark 2.1 There is no assumption on in the above result. Since, for any ,
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.
Theorem 2.2 Let be a locally square integrable martingale. Then, for all , , ,
Moreover, we also have the result
Remark 2.2 Since, for any , , and ,
the inequality (4) is better than the inequality (2). Similarly, we have
Hence, Theorem 2.2 improves Theorem 1.2.
From Lemma 2.2 in Section 2, we know that and satisfy the canonical assumption, i.e.
Therefore, we can obtain the following theorem.
Theorem 2.3 Let be a martingale different sequence with respect to the filtration and suppose that for all . Then, for ,
and for ,
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.
In [, Theorem 3.1], De la Peña and Pang obtained the following inequality: Let T be any stopping time with respect the filtration and assume almost surely. Then, for all ,
By comparing the inequalities (6) and (7), we know that the inequality (6) is better than the inequality (7). More precisely, we have
The following result may be of independent interest.
Theorem 2.4 Let be a pair of random variables with in the probability space satisfying the canonical assumption. For every , and ,
From Theorem 2.4, we have the following.
Corollary 2.1 Let be a locally square integrable martingale. Then, for every , and , we have
2.2 Proofs of main results
We start with the following basic lemma.
Lemma 2.1 Let X be a square integrable random variable with and . Then, for all , we have
Proof First, we shall prove
then, by a straightforward calculation, we have
it follows that
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.
Lemma 2.2 Let be a locally square integrable martingale. For all and , denote
Then, for all , is a positive supermartingale with .
Proof For all and , we have
where , and . Hence, we deduce from Lemma 2.1 that, for all ,
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 .
Proof of Theorem 2.1 First of all, according to the condition of Lemma 2.2, we denote
For all let . We define where and . By Markov’s inequality, we have, for all ,
Hence, we deduce from Lemma 2.2 that, for all ,
Dividing both sides of (12) by and choosing the value , we find that
We also have the same upper bound for , immediately leading to the result of (3). □
Proof of Theorem 2.2 We are going to list the proof of Theorem 2.2 in the special case and . For all , let
By Cauchy-Schwarz’s inequality, we have, for all ,
where is defined in (11). Consequently, we obtain from (13) with the particular choice that
Therefore, if we divide both sides of (14) by , we find that
The same upper bound holds for , which clearly implies (4). Furthermore, for all , let
By Hölder’s inequality, we have, for all and ,
Consequently, as , we can deduce from (15) and the particular choice that
We also find the same upper bound for , which completes the proof of Theorem 2.2. □
Proof of Theorem 2.3 Because is a martingale different sequence, it satisfies the canonical assumption, i.e. . Putting
then, according to the canonical assumption and Fubini’s theorem, for any , we have
For any measurable set , by Markov’s inequality, we get
By using Hölder’s inequality, we get
Let , and by using Hölder’s inequality, we have the following inequality:
Hence, from (16), it follows that
Now letting , we will have
So we can get the inequality
Proof of Theorem 2.4 Given , let and define random events , , where K denotes the integer part of . Since the pair satisfies the canonical assumption, we have
Therefore, we conclude that
We need to choose such that the bound in (17) is possibly small. Now, by taking , we have
Since for , then we have , which, together with (18), yields the desired result. □
3 Linear regressions
In this section, let us consider the deviation inequality of the least-squares estimate of the unknown parameter in linear regressive model. For all , let
where , , and are the observation, the regression variable, and the driven noise, respectively. Suppose that is a sequence of independent and identically distributed random variables, and is a sequence of identically distributed random variables with mean zero and variance . Furthermore, assume that is independent of where . The least-squares estimate of the unknown parameter θ is given by
which yields from (19)
Let H and L be the cumulant generating functions of the sequence and , respectively given, for all , by
Bercu and Touati  obtained the following result.
Corollary 3.1 [, Corollary 5.1]
Assume that L is finite on some interval with and denote by I its Fenchel-Legendre transform on ,
Then, for all , and , we have
Now, we give the following theorem.
Theorem 3.1 Under the conditions of Corollary 3.1, for all , , and , we have
Remark 3.1 Obviously, the upper bound in (22) is better than the bound (21).
Proof From (20), for all , and , we get
By using the inequality (5), it follows that
Furthermore, since, for any ,
then, from the above discussions, the desired results can be obtained. □
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This work is supported by IRTSTHN (14IRTSTHN023), NSFC (No. 11001077), NCET (NCET-11-0945), and Plan For Scientific Innovation Talent of Henan Province (124100510014).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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Cite this article
Chen, S., Wang, Z., Xu, W. et al. Exponential inequalities for self-normalized martingales. J Inequal Appl 2014, 289 (2014). https://doi.org/10.1186/1029-242X-2014-289
- exponential inequalities
- self-normalized martingale difference sequence
- random variables
- tail probabilities