Exponential inequalities for self-normalized processes with applications

We prove the following exponential inequality for a pair of random variables $(A,B)$ with $B >0$ satisfying the canonical assumption , $E[\exp(\lambda A - \frac{\lambda^2}{2} B^2)]\leq 1$ for $\lambda \in R$, $$P\left( \frac{|A|}{\sqrt{ \frac{2q-1}{q} \left(B^2+ (E[|A|^p])^{2/p} \right) }} \geq x \right) \leq \left(\frac{q}{2q-1} \right)^{\frac{q}{2q-1}} x^{-\frac{q}{2q-1}} e^{-x^2/2} $$ for $x>0$, where $1/p+ 1/q =1$ and $p\geq1$. Applying this inequality, we obtain exponential bounds for the tail probabilities for self-normalized martingale difference sequences. We propose a method of hypothesis testing for the $L^p$-norm $(p \geq 1)$ of $A$ (in particular, martingales) and some stopping times. We apply this inequality to the stochastic TSP in $[0,1]^d$ ($d\geq 2$), connected to the CLT.


Introduction
Self-normalized stochastic processes are frequently found in statistical applications. They have the property of (in the standard form) being unit free and frequently eliminate or weaken moment assumptions. The prototypical example of a self-normalized process is Student's t-statistic, which is used in statistical analysis to test if the mean of a normally distributed sample has a value specified in a null hypothesis when the standard deviation of the underlying distribution is unknown. Let {X i : i ≥ 1} be a sequence of i.i.d. normal random variables with mean 0 and variance σ 2 . The 372 DOI: 10.1214/ECP.v14-1490 sample meanX n = n i=1 X i /n and the sample variance s 2 n = n i=1 (X i −X n ) 2 /(n−1). The t-statistic T n = nX n /s n has the Student t-distribution of freedom n − 1, which converges to a standard normal random variable as n → ∞. Some limit theorems and moment bounds have been proved for the t-statistic by observing that T n is a function of self-normalized sums: , where S n = n i=1 X i and V 2 n = n i=1 X 2 i . The limit distribution of the self-normalized sums S n /V n has been proved by Efron (1969) and Logan et al. (1973). Giné, Götze and Mason (1997) prove that T n has a limiting standard normal distribution if and only if X 1 is in the domain of attraction of a normal law by making use of exponential and L p bounds for the self-normalized sums S n /V n . Since the 1990s, there have been active developments of the probability theory of self-normalized processes. We refer to de la Peña, Lai and Shao (2009) for the comprehensive review of the state of the art of the theory and its applications in statistical inference. Here we make a contribution to this theory by proving a new exponential inequality for self-normalized processes (Theorem 2.1). We start by considering a pair of random variables (A, B) with B > 0 satisfying the following By the method of mixtures, de la Peña, Klass and Lai (2004) prove the following exponential bound for such a pair (A, B), It is connected to the central limit theorem (CLT) and provides related control on the tail. Here we will prove a new exponential inequality for |A|/ ((2q − 1)/q) [ The new inequality is presented and proved in §2. Then, it is applied to obtain an exponential upper bound for the tail probability for self-normalized martingale difference sequences in §3. We propose a method of hypothesis testing for L p -norms (p ≥ 1) for martingales and stopping times in §4. In §5, we present the new inequality for the stochastic TSP.

Exponential Inequalities
In this section, we present and prove the new exponential inequality for self-normalized processes. . Suppose E[|A| p ] < ∞ for some p ≥ 1. Then for any x > 0 and for q ≥ 1 such that 1/p + 1/q = 1,  (2004),

) to obtain an exponential bound without knowing the distribution of B or its moments. This is because, by Theorem 3.3 in de la Peña, Klass and Lai
where x 0 is some large constant, and Example 2.6 in de la Peña, Klass and Lai (2007) shows that this upper bound is sharp.
Proof. First of all, we establish the following identity, for any C > 0, This holds because by the canonical assumption and Fubini's Theorem, Let G ∈ by any measurable set. Then, by Markov's inequality, Now, by Hölder's inequality, we can choose C = (E[|A| p ]) 2/p so that for p and q satisfying 1/p +1/q = 1, again by Hölder's inequality, We can take limit on both sides of (2.1) by the monotone convergence theorem as q → ∞ (or p → 1) to obtain (2.3).

Inequalities for Martingale Difference Sequences
Bercu and Touati (2008) prove that martingale difference sequences satisfy the canonical assumption in the form of the following lemma.
By Lemma 3.1 and Theorem 2.1, we obtain the following theorem for martingale difference sequences. In time series analysis, Theorem 3.1 can also be used to establish useful bounds for moving average sequences since they can be regarded as martingale difference sequences.
and for x > 0, Proof. We first consider the sequences T ∧n i=1 X i , which can be written as Then (

Lemma 8.8 in de la Peña, Lai and Shao (2009) says that for such a sequence {X
Our inequality (3.2) provides a tighter upper bound than the inequality (3.5) for large x.

Remark 3.3. Lemma A.4 in de la Peña, Klass and Lai (2007) proves that for a sequence {X i : i ≥ 1} of conditionally symmetric random variables adapted to the filtration
By Theorem 2.1, we obtain for x > 0,

Applications to Hypothesis Testing
With the new inequality (2.1), we propose a method to test the L p -norm (p ≥ 1) for a random variable A. We first choose another positive random variable B such that the pair (A, B) where x α is such that As a special case when p = 2, we can test the variance of A with mean 0. A useful application is to test if the L p -norms of martingales are equal to some specific values.

Application to the Stochastic TSP
In the stochastic modeling of the TSP, let X 1 , ..., X n be i.i.d. uniformly distributed on [0, 1] d (d ≥ 2) and T n be the shortest closed path through the n random points X 1 , ..., X n . It has been shown that the deviation of T n from its mean E[T n ] is remarkably small, see Steele (1981) and Rhee and Talagrand (1987, 1989a, 1989b. In particular, by Azuma's inequality, we have for n ≥ 2, for 1 ≤ i ≤ n, and i = σ{X 1 , ..., X i }, the σ-algebra generated by X 1 , ..., X i .
Proof. First, since T n is n measurable, we can write where {d k : 1 ≤ k ≤ n} is a martingale difference sequence. It can be easily checked that for all 1 ≤ i ≤ n, where || · || ∞ is the essential supremum norm; see (2.8) in Steele (1997) and Corollary 5 in Rhee and Talagrand (1987). So E[d 2 i ] < ∞ for all 1 ≤ i ≤ n and the conditions in Lemma 3.1 are satisfied. Then, by Theorem 3.1, we have that for t > 0,   This inequality is related to the inequality (3.5) in Rhee and Talagrand (1987).

Remark 5.2.
In the proof of Theorem 5.1, by (5.3), we have Then, for t > 0,