Consider a linear random field $ X=\{X_{j, k}; (j, k)\in\mathbb Z^2\} $ defined on a probability space $ (\Omega, \mathcal F, \mathbb P) $ by
where $ \{a_{r, s}; (r, s)\in \mathbb Z^2\} $ is a sequence of constants, the innovations $ \{\xi_{r, s}; (r, s)\in\mathbb Z^2\} $ are independent and identically distributed (i.i.d.) centered square integrable random variables, with common law $ \mathcal L (\xi_{0, 0})=\mu $, and the variance Var $ (\xi_{0, 0})=1 $. Assume that
Under the condition (1.2), $ X_{j, k} $ in (1.1) is well defined, see Lemma A.1 in [1]. The family $ \{X_{j, k}; (j, k)\in \mathbb Z^2\} $ is strictly stationary with the spectral density given by
where $ {\mathit{\boldsymbol{ u }}}\cdot {\mathit{\boldsymbol{ t }}}:=u_1t_1+u_2t_2 $ for any $ {\mathit{\boldsymbol{ u }}}=(u_1, u_2)\in \mathbb Z^2 $ and $ {\mathit{\boldsymbol{ t }}}=(t_1, t_2)\in {\mathit{\boldsymbol{ I }}} $, $ {\mathit{\boldsymbol{ I }}} $ is the torus identified with $ [-\pi, \pi)^2 $ in the usual way.
The empirical periodogram is one of the main tools in the estimation of the unknown spectral density $ f $. It is defined by
where $ {\mathit{\boldsymbol{ t }}}=(t_1, t_2)\in {\mathit{\boldsymbol{ I }}} $, $ {\mathit{\boldsymbol{ n }}}=(n, n) $, $ {1\mathit{\boldsymbol{}}}\le {\mathit{\boldsymbol{ u }}}\le {\mathit{\boldsymbol{ n }}} $ means that $ 1\le u_1\le n $, $ 1\le u_2\le n $ for $ {\mathit{\boldsymbol{ u }}}=(u_1, u_2)\in \mathbb Z^2 $.
The limit theorems of the empirical periodogram $ \mathcal I_{{\mathit{\boldsymbol{ n }}}} $ usually involve the following quadratic forms
There have been abundant literatures contributing to the study of limit theorems for $ \mathcal I_{{\mathit{\boldsymbol{ n }}}} $ and $ \Phi_{{\mathit{\boldsymbol{ n }}}} $, we refer the reader to Rosenblatt [2] and references therein.
The main purpose of this paper is to estalish the moderate deviation principle(MDP in short) for the empirical periodogram $ \mathcal I_{{\mathit{\boldsymbol{ n }}}} $ and quadratic forms $ \Phi_{{\mathit{\boldsymbol{ n }}}} $. In order to do these, we put forward sone conditions such as $ L^q({\mathit{\boldsymbol{ I }}}, d{\mathit{\boldsymbol{ t }}}) $-integrability of the spectral density $ f $ and a Logarithmic Sobolev Inequality (LSI in short) for the law $ \mu $ of the driving random variable $ \xi $. Moreover, as statistical applications, we provide the moderate deviation estimates of the least square and the Yule-Walker estimators for unilateral autoregression stationary fields.
This paper is organized as follows. In section 2, we establish the MDP for quadratic form and the empirical periodogram, and give a statistical application based on our MDP results. Some lemmas of dependent stationary fields and the proofs of the main results are given in section 3.
Assume the following conditions:
(H1) The law $ \mu $ of the driving random variable $ \xi $ satisfies a LSI, i.e., there exists a constant $ C>0 $ such that
for every smooth $ h $ such that $ \mathbb E_{\mu}(h^2\log^{+} h^2)<\infty $, where
(H2) The spectral density function $ f $ is in $ L^q({\mathit{\boldsymbol{ I }}}, d{\mathit{\boldsymbol{ t }}}) $, where $ 2<q\le +\infty $.
(H3) The sequence of positive numbers $ \{b_n\}_{n\in \mathbb N} $ satisfies that $ b_n\rightarrow \infty $ and $ b_n n^{1/q-1/2}\rightarrow 0 $ as $ n\rightarrow \infty. $ Here $ q $ is the constant appearing in (H2).
Remark The LSI (2.1) implies that there exists some constant $ \delta>0 $ such that
See [3, Chapter 3] for more details on the LSI.
Let
be the cumulant of order $ 4 $ of the random variable $ \xi_{{\mathit{\boldsymbol{0}}}} $.
Theorem 2.1 Assume (H1)-(H3) hold. Then for every $ \lambda=(\lambda_{{\mathit{\boldsymbol{ k }}}})_{{\mathit{\boldsymbol{0}}}\le {\mathit{\boldsymbol{ k }}}\le {\mathit{\boldsymbol{ m }}}}\in \mathbb R^{(m+1)^2} $,
where $ \Sigma^2_{{\mathit{\boldsymbol{ k }}}, {\mathit{\boldsymbol{ l }}}}, {\mathit{\boldsymbol{0}}}\le {\mathit{\boldsymbol{ k }}}, {\mathit{\boldsymbol{ l }}}\le {\mathit{\boldsymbol{ m }}} $ is given by
In particular,
satisfies the LDP on $ \mathbb R^{(m+1)^2} $ with speed $ b_n^4 $ and with the rate function given by
As a consequence of Theorem 2.1, we have the following MDP by the contraction principle ([4, Theorem 4.2.1]).
Corollary 2.2 Assume (H1)-(H3) hold. For all $ {\mathit{\boldsymbol{ l }}}\ge {\mathit{\boldsymbol{0}}} $, $ \frac{1}{nb_{n}^2 } \sum\limits_{{1\mathit{\boldsymbol{}}}\le {\mathit{\boldsymbol{ k }}}\le {\mathit{\boldsymbol{ n }}}} (X_{{\mathit{\boldsymbol{ k }}}} X_{{\mathit{\boldsymbol{ k }}}+{\mathit{\boldsymbol{ l }}}}- \mathbb EX_{{\mathit{\boldsymbol{ k }}}}X_{{\mathit{\boldsymbol{ k }}}+{\mathit{\boldsymbol{ l }}}}) $ satisfies the LDP on $ \mathbb R $ with speed $ b_n^4 $ and with the rate function given by
with the convention that $ a/0=+\infty $ for all $ a>0 $ and $ 0/0:=0 $.
From Theorem 2.1 and the projective limit method (see [4]), we obtain the functional type's MDP for
Theorem 2.3 Assume (H1)-(H3) hold. Let $ 1\le p<2 $ and $ p'\in[2, +\infty] $ the conjugated number, i.e., $ 1/p+1/p'=1 $. Assume that
Then $ (\mathcal L_{n})_{n\ge1} $ satisfies the LDP on $ (L^p({\mathit{\boldsymbol{ I }}}, d{\mathit{\boldsymbol{ t }}}), \sigma(L^p({\mathit{\boldsymbol{ I }}}, d{\mathit{\boldsymbol{ t }}}), L^{p'}({\mathit{\boldsymbol{ I }}}, d{\mathit{\boldsymbol{ t }}})) ) $ with speed $ b_n^4 $ and with the rate function given by
where $ \eta $ is even means that $ \eta({\mathit{\boldsymbol{ t }}})=\eta(-{\mathit{\boldsymbol{ t }}}), \forall {\mathit{\boldsymbol{ t }}}\in {\mathit{\boldsymbol{ I }}} $.
As a consequence of Theorem 2.3, we have the following MDP.
Corollary 2.4 Under the assumptions of Theorem 2.3, we have that for all $ h\in L^{p'}({\mathit{\boldsymbol{ I }}}, d{\mathit{\boldsymbol{ t }}}) $,
where
with $ \tilde h({\mathit{\boldsymbol{ t }}})=(h({\mathit{\boldsymbol{ t }}})+h(-{\mathit{\boldsymbol{ t }}}))/2 $. In particular, $ \frac{1}{(2\pi)^2}\int_{{\mathit{\boldsymbol{ I }}}} h({\mathit{\boldsymbol{ t }}})\mathcal L_n({\mathit{\boldsymbol{ t }}})d{\mathit{\boldsymbol{ t }}} $ satisfies the LDP on $ \mathbb R $ with speed $ b_n^4 $ and with the rate function given by $ I_h(z):=\frac{z^2}{2\sigma^2(h)}. $
Consider the unilateral autoregression process:
where the sequence $ \{\varepsilon_{s, t}\}_{s, t\in \mathbb Z} $ is i.i.d. with common law $ \mu $, satisfying a LSI, and $ \mathbb E (\varepsilon_{s, t})=0, \mathbb E(\varepsilon_{s, t}^2)=1 $ and $ \rho\in(-1, 1) $ is the unknown parameter. Assume that the sequence $ \{X_{0, k}\}_{k\in \mathbb N} $ is i.i.d. with common law of $ \sum_{k=0}^{\infty}\rho^k\varepsilon_{-k, 0} $, which is independent of $ \{\varepsilon_{s, t}\}_{s, t\ge1} $. $ \{ X_{s, t}\}_{s, t\in \mathbb N} $ is thus a centered stationary field, which is a special one of the unilateral autoregression fields, see [5] and [6].
Let $ \hat \rho_n $ be the least square estimator of $ \rho $ given by
and let $ \tilde \rho_n $ be the Yule-Walker estimator of $ \rho $ given by
It is well-known that $ \hat\rho_n $ and $ \tilde \rho $ are consistent and asymptotically normal, see [5] and [6]. Moreover, Bercu et al.[7] showed that the LDP of the Yule-Walker estimator is better than the one of the least square estimator.
By using Theorem 2.1 and the same proof of Proposition 3.1 in [8], one can prove the following MDP of the least square and the Yule-Walker estimator. These two estimators share the same MDP. Here we omit the proof.
Proposition 2.5 Let $ (b_n) $ be a sequence of positive numbers satisfying that $ b_n\rightarrow \infty $ and $ b_n n^{1/q-1/2}\rightarrow 0 $ as $ n\rightarrow \infty. $ Then $ \frac{n}{b_n^2}(\hat\rho_n-\rho) $ (or $ \frac{n}{b_n^2}(\tilde\rho_n-\rho) $) satisfies the LDP on $ \mathbb R $ with speed $ b_n^4 $ and with the rate function given by
The proof of Theorem 2.1 and Theorem 2.3 are similar to that in [8]. Here, we only give the sketch of the proof for Theorem 2.1. First, we give some lemmas.
In this part, we give several facts concerning the Toeplitz matrix, the Fejèr approximation and the MDP of $ m $-dependent stationary fields.
For an $ n\times n $ matrix $ A $, denote the usual operator norm $ \|A\|=\sup\limits_{x\in \mathbb R^n} \frac{|Ax|}{|x|} $. For any even function $ h\in L^1({\mathit{\boldsymbol{ I }}}, d{\mathit{\boldsymbol{ t }}}) $, $ T_{{\mathit{\boldsymbol{ n }}}}(h)=(\hat r_{{\mathit{\boldsymbol{ k }}}-{\mathit{\boldsymbol{ l }}}}(h))_{{1\mathit{\boldsymbol{}}}\le {\mathit{\boldsymbol{ k }}}, {\mathit{\boldsymbol{ l }}} \le {\mathit{\boldsymbol{ n }}}} $ is the Toeplitz matrix associated with $ h $, where $ \hat r_{{\mathit{\boldsymbol{ k }}}}(h) $ is the $ {\mathit{\boldsymbol{ k }}} $th Fourier coefficient of $ h $ given by
Lemma 3.1 [9, Lemma 1] If $ f\in L^q({\mathit{\boldsymbol{ I }}}, d{\mathit{\boldsymbol{ t }}}) $, where $ 1\le q\le \infty $, then for all $ {\mathit{\boldsymbol{ n }}}\ge \bm1 $, we have
Lemma 3.2 [9, Theorem 1] Let $ f_{k}\in L^{q_k}({\mathit{\boldsymbol{ I }}}, d{\mathit{\boldsymbol{ t }}}) $ with $ q_{k}\ge1 $ for $ k=1, \cdots, p $ and $ \sum\limits_{k=1}^p (1/q_k)\le 1 $. Then
Let $ m $ be a given positive integer, a random field $ (Z_{{\mathit{\boldsymbol{ n }}}})_{{\mathit{\boldsymbol{ n }}} \in \mathbb Z^2} $ is called $ m $-dependent, if for any finite sets $ A, B\subset \mathbb Z^2 $ with $ d(A, B)>m $, the $ \sigma $-algebras $ \sigma\{Z_{k, l}; (k, l)\in A\} $ and $ \sigma\{Z_{k, l}; (k, l)\in B\} $ are independent. Here
See [10] for the study of the strong large number law for $ m $-dependent random fields.
The following Lemma is the generalization of the MDP result for the $ m $-dependent stationary processes in [11].
Lemma 3.3 Let $ (Z_{{\mathit{\boldsymbol{ n }}}})_{{\mathit{\boldsymbol{ n }}}\in \mathbb Z^2} $ be a centered $ m $-dependent stationary random field in $ \mathbb R^N $, such that
Then for all $ \lambda\in \mathbb R^N $,
Proof The proof is inspired by the method in the proof of [11, Theorem 1.1]. Here, we give the sketch of the proof. First, we prove this lemma in the case $ m=1 $. Fix the integer $ p>1 $ and for each $ n\ge1 $, we write $ n=k_np+r_n, $ where $ k_n $ and $ r_n $ are non-negative integers with $ 0\le r_n\le p-1 $. Define
Then, $ \{Y_{r, s}\}_{(r, s)\in \mathbb N^2} $ is an independent and identically distributed sequence. By the moderate deviation principle for i.i.d. random variables (see [4, Theorem 3.7.1]), we have
Similarly,
By (H3) and (3.2), we have
Let $ S_{{\mathit{\boldsymbol{ n }}}}=\sum\limits_{{1\mathit{\boldsymbol{}}}\le {\mathit{\boldsymbol{ k }}}\le {\mathit{\boldsymbol{ n }}}}Z_{{\mathit{\boldsymbol{ k }}}}, $ noticing that
Then for any $ a_1, a_2, a_3>1 $ with $ 1/a_1+1/a_2+1/a_3=1 $, by the Hölder inequality, we have
By (3.4)–(3.6), we know that
Firstly letting $ p\rightarrow \infty $ and then letting $ a_1\rightarrow 1 $ in (3.9), we have
Similarly, from (3.7),
This, together with the previous calculus, implies
Hence, we have (3.3).
For general $ m>1 $, (3.3) can be proved by the argument in the proof of Theorem 1.1 of [11]. We omit the details here.
The proof is complete.
First, we approximate the linear random field by a linear random field of finite range $ 2N $, which satisfies the MDP.
Let $ X_{j, k}^N $ be the Fejér approximation of $ X_{j, k} $, that is
where $ a_{r, s}^N=a_{r, s}\left(1-\frac{|r|}{N}\right)\left(1-\frac{|s|}{N}\right)_{|r|\le N, |s|\le N} $.
Set
Then the family $ \left\{(X^N_{{\mathit{\boldsymbol{ k }}}} X^N_{{\mathit{\boldsymbol{ k }}}+{\mathit{\boldsymbol{ l }}}})_{{\mathit{\boldsymbol{0}}}\le {{\mathit{\boldsymbol{ l }}}}\le {{\mathit{\boldsymbol{ m }}}}}\in \mathbb R^{(m+1)^2}, {\mathit{\boldsymbol{ k }}}\in \mathbb Z^2\right\} $ is a $ 2N $-dependent stationary field. By (2.3), for all $ N\ge1 $, there exists a constant $ \eta>0 $ such that $ \mathbb E \big(e^{\eta |X^N_{{\mathit{\boldsymbol{ k }}}} X^N_{{\mathit{\boldsymbol{ k }}}+{\mathit{\boldsymbol{ l }}}}|} \big)<\infty $.
By Lemma 3.3, we get that for any $ N $ fixed, and for all $ \lambda\in \mathbb R^{(m+1)^2} $,
where $ \Sigma^{2, N} $ is the covariance matrix given in Lemma 3.3. Then, by the Gärtner-Ellis theorem, we know that $ Q_n^N $ satisfies the MDP on $ \mathbb R^{(m+1)^2} $ with the good rate function $ I_N({\mathit{\boldsymbol{ x }}})=\sup\limits_{\lambda\in \mathbb R^{(m+1)^2}}\left\{\langle \lambda, {\mathit{\boldsymbol{ x }}}\rangle -\frac{1}{2}\langle \lambda, \Sigma^{2, N}\lambda\rangle \right\}. $ Furthermore, by [2, Theorem 4.6.2], $ \Sigma^{2, N}_{{\mathit{\boldsymbol{ k }}}, {\mathit{\boldsymbol{ l }}}} $ can be expressed as (2.5) with $ f $ replaced by $ f^N $, where $ f^N({\mathit{\boldsymbol{ t }}})= |\sum\limits_{{\mathit{\boldsymbol{ u }}}\in \mathbb Z^2}a_{{\mathit{\boldsymbol{ u }}}}^N e^{i {\mathit{\boldsymbol{ u }}}\cdot {\mathit{\boldsymbol{ t }}}}|^2. $
Then we should show that the approximation is a good one in the sense of the MDP, and we establish the convergence of the rate function. These proofs are the same as the proof of Theorem 2.1 in [8] by using the techniques of LSI. We omit the details here.
2023, Vol. 43 Issue (2): 126-134
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2023, Vol. 43 Issue (2): 126-134 PDF