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Slepian’s inequality for Gaussian processes with respect to weak majorization
Journal of Inequalities and Applications volume 2013, Article number: 5 (2013)
Abstract
In this paper, we obtain some sufficient conditions for Slepian’s inequality for Gaussian processes with respect to weak majorization. For our results, we also provide an application example.
MSC:60E15, 62G30.
1 Introduction and main results
Gaussian processes are natural extensions of multivariate Gaussian random variables to infinite (countably or continuous) index sets. For Gaussian processes, strong and weak stationarity are the same concept. Gaussian processes are by far the most accessible and wellunderstood processes (on uncountable index sets), which are important in statistical modeling because of properties inherited from the normal one, and many deep theoretical analyses of various properties are available.
Let X=({X}_{1},\dots ,{X}_{n}) and {X}^{\ast}=({X}_{1}^{\ast},\dots ,{X}_{n}^{\ast}) be two centered Gaussian random vectors with covariance matrices \mathrm{\Sigma}=({\sigma}_{ij}) and {\mathrm{\Sigma}}^{\ast}=({\sigma}_{ij}^{\ast}), respectively. The wellknown Slepian inequality [1] states that if {\sigma}_{ii}={\sigma}_{ii}^{\ast} and {\sigma}_{ij}\le {\sigma}_{ij}^{\ast} for every i,j=1,\dots ,n, then for any x\in R,
Slepian’s inequality and its modifications are an essential ingredient in the proofs of many results being concerned with sample path properties of Gaussian processes. See, e.g., Adler and Taylor [2] and Maurer [3]. Some sufficient conditions for Slepian’s inequality with respect to majorization for two Gaussian random vectors have been given in Fang and Zhang [4].
Majorization is a preordering on vectors by sorting all components in nonincreasing order, which is a very interesting topic in various fields of mathematics and statistics. The history of investigating majorization should date back to Schur [5] and Hardy et al. [6]. The reader can find that majorization has been connected with combinatorics, analytic inequalities, numerical analysis, matrix theory, probability and statistics in Marshall and Olkin [7]. Recent research on majorization with respect to matrix inequalities and norm inequalities has been carried out by Ando [8].
In this paper, we establish four Slepian’s inequalities for Gaussian processes with respect to weak majorization, with their proofs and an application given in Section 2. Firstly, we recall the definitions of majorization and weak majorization.
Definition 1.1 (Marshall and Olkin [7])
Let \mathit{\lambda}=({\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n}), {\mathit{\lambda}}^{\ast}=({\lambda}_{1}^{\ast},{\lambda}_{2}^{\ast},\dots ,{\lambda}_{n}^{\ast}) denote two ndimensional real vectors. Let {\lambda}_{[1]}\ge {\lambda}_{[2]}\ge \cdots \ge {\lambda}_{[n]} and {\lambda}_{[1]}^{\ast}\ge {\lambda}_{[2]}^{\ast}\ge \cdots \ge {\lambda}_{[n]}^{\ast} denote the components of λ and {\mathit{\lambda}}^{\ast} in decreasing order respectively. Similarly, let {\lambda}_{(1)}\le {\lambda}_{(2)}\le \cdots \le {\lambda}_{(n)} and {\lambda}_{(1)}^{\ast}\le {\lambda}_{(2)}^{\ast}\le \cdots \le {\lambda}_{(n)}^{\ast} denote the components of λ and {\mathit{\lambda}}^{\ast} in increasing order respectively.

(1)
{\mathit{\lambda}}^{\ast} is said to be majorized by λ, in symbols \mathit{\lambda}{\u2ab0}_{m}{\mathit{\lambda}}^{\ast}, if
\sum _{i=1}^{m}{\lambda}_{[i]}\ge \sum _{i=1}^{m}{\lambda}_{[i]}^{\ast}
for m=1,2,\dots ,n1, and {\sum}_{i=1}^{n}{\lambda}_{i}={\sum}_{i=1}^{n}{\lambda}_{i}^{\ast}.

(2)
{\mathit{\lambda}}^{\ast} is said to be weak lower majorized by λ, in symbols \mathit{\lambda}{\u2ab0}_{w}{\mathit{\lambda}}^{\ast}, if
\sum _{i=1}^{m}{\lambda}_{[i]}\ge \sum _{i=1}^{m}{\lambda}_{[i]}^{\ast}
for m=1,2,\dots ,n1, and {\sum}_{i=1}^{n}{\lambda}_{i}\ge {\sum}_{i=1}^{n}{\lambda}_{i}^{\ast}.

(3)
{\mathit{\lambda}}^{\ast} is said to be weak upper majorized by λ, in symbols \mathit{\lambda}{\u2ab0}^{w}{\mathit{\lambda}}^{\ast}, if
\sum _{i=1}^{m}{\lambda}_{(i)}\le \sum _{i=1}^{m}{\lambda}_{(i)}^{\ast}
for m=1,2,\dots ,n1, and {\sum}_{i=1}^{n}{\lambda}_{i}\le {\sum}_{i=1}^{n}{\lambda}_{i}^{\ast}.
The main results of the paper are stated as follows.
Theorem 1.2 Let X(t) and {X}^{\ast}(t) be separable Gaussian processes where t\in [0,T]. We assume that the two processes have the same covariance function, i.e.,
for all s,t\in [0,T]. Denote u={inf}_{t\in [0,T]}\{EX(t),E{X}^{\ast}(t)\}, v={sup}_{t\in [0,T]}\{EX(t),E{X}^{\ast}(t)\}. Let 0\le {t}_{1}\le {t}_{2}\le \cdots \le {t}_{n}\le T be a sequence of arbitrary partitions of [0,T]. Let f:[u,v]\to R be a strictly monotone function, and denote {\mathit{\mu}}_{f}=(f(EX({t}_{1})),\dots ,f(EX({t}_{n}))), {\mathit{\mu}}_{f}^{\ast}=(f(E{X}^{\ast}({t}_{1})),\dots ,f(E{X}^{\ast}({t}_{n}))).

(1)
If {f}^{\prime}(y)>0, {f}^{\u2033}(y)\ge 0 for all y\in [u,v], and {\mathit{\mu}}_{f}{\u2ab0}^{w}{\mathit{\mu}}_{f}^{\ast}, then
P(\underset{t\in [0,T]}{inf}X(t)\ge x)\le P(\underset{t\in [0,T]}{inf}{X}^{\ast}(t)\ge x)
for all x\in R;

(2)
If {f}^{\prime}(y)<0, {f}^{\u2033}(y)\le 0 for all y\in [u,v], and {\mathit{\mu}}_{f}{\u2ab0}_{w}{\mathit{\mu}}_{f}^{\ast}, then
P(\underset{t\in [0,T]}{inf}X(t)\ge x)\le P(\underset{t\in [0,T]}{inf}{X}^{\ast}(t)\ge x)
for all x\in R;

(3)
If {f}^{\prime}(y)>0, {f}^{\u2033}(y)\le 0 for all y\in [u,v], and {\mathit{\mu}}_{f}{\u2ab0}_{w}{\mathit{\mu}}_{f}^{\ast}, then
P(\underset{t\in [0,T]}{sup}X(t)\le x)\le P(\underset{t\in [0,T]}{sup}{X}^{\ast}(t)\le x)
for all x\in R;

(4)
If {f}^{\prime}(y)<0, {f}^{\u2033}(y)\ge 0 for all y\in [u,v], and {\mathit{\mu}}_{f}{\u2ab0}^{w}{\mathit{\mu}}_{f}^{\ast}, then
P(\underset{t\in [0,T]}{sup}X(t)\le x)\le P(\underset{t\in [0,T]}{sup}{X}^{\ast}(t)\le x)
for all x\in R.
In Theorem 1.2, after setting f(x)=x, we can easily get the following result.
Corollary 1.3 Under the same conditions on X(t), {X}^{\ast}(t) and \{{t}_{i},1\le i\le n\} as in Theorem 1.2, the following statements hold.

(1)
If (EX({t}_{1}),\dots ,EX({t}_{n})){\u2ab0}^{w}(E{X}^{\ast}({t}_{1}),\dots ,E{X}^{\ast}({t}_{n})), then
P(\underset{t\in [0,T]}{inf}X(t)\ge x)\le P(\underset{t\in [0,T]}{inf}{X}^{\ast}(t)\ge x)
for all x\in R;

(2)
If (EX({t}_{1}),\dots ,EX({t}_{n})){\u2ab0}_{w}(E{X}^{\ast}({t}_{1}),\dots ,E{X}^{\ast}({t}_{n})), then
P(\underset{t\in [0,T]}{sup}X(t)\le x)\le P(\underset{t\in [0,T]}{sup}{X}^{\ast}(t)\le x)
for all x\in R.
2 Proof and application
Proof of Theorem 1.2 Each of the four conclusions in Theorem 1.2 can be proved by the similar ideas. So, we only give the detailed proof of part (3) here. Let 0\le {t}_{1}\le {t}_{2}\le \cdots \le {t}_{n}\le T be a sequence of partitions of [0,T], and \tau ={max}_{1\le i\le n}\mathrm{\u25b3}{t}_{i}, so we can obtain Gaussian random variables X({t}_{1}),\dots ,X({t}_{n}) and {X}^{\ast}({t}_{1}),\dots ,{X}^{\ast}({t}_{n}), respectively. Thus
By using the conditions of Theorem 1.2, we know
for all i,j=1,\dots ,n. And
From Fang and Zhang [4], we have
Since
and
According to the above three expressions, we have
□
An application
Let X(t)={t}^{2}+{B}^{1,H,K}(t) and {X}^{\ast}(t)={t}^{3}+{B}^{2,H,K}(t) be Gaussian processes, where {B}^{i,H,K}(t), i=1,2, H\in (0,1), K\in (0,1], are centered Gaussian processes such that
for all s,t\in [0,1].
It is easy to check that X(t) and {X}^{\ast}(t) satisfy the conditions in Theorem 1.2.
Let 0\le {t}_{1}\le {t}_{2}\le \cdots \le {t}_{n}\le 1 be a sequence of partitions of [0,1], then
From Corollary 1.3, we have, for all x\in R,
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Acknowledgements
This research is supported by the National Statistical Science Research Project of China (No. 2012LY158), Natural Science Foundation of AnhuiProvince (No. 1208085MA11).
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Fang, L. Slepian’s inequality for Gaussian processes with respect to weak majorization. J Inequal Appl 2013, 5 (2013). https://doi.org/10.1186/1029242X20135
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DOI: https://doi.org/10.1186/1029242X20135