1 Introduction
This paper is devoted to investigation of the supremum of averaged deviations of stochastic processes from Orlicz spaces of random variables using the method of majorizing measures. In particular, we estimate functionals of the following type:
\[\underset{t\in \mathbb{T}}{\sup }\left|X(t)-f(t)-\frac{1}{\mu (\mathbb{T})}\int _{\mathbb{T}}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|\]
where $(\mathbb{T},\mathcal{B},\mu )$ is a measurable space with finite measure $\mu (\mathbb{T})<\infty $, and $f(u)$ is some function. In particular, using the obtained with probability one estimates for such a functional, we are able to estimate the distribution of $\sup _{t\in \mathbb{T}}|X(t)-f(t)|$. A special attention is devoted to the Orlicz spaces such as the $L_{q}$ spaces.The method of majorizing measures is used in the theory of Gaussian stochastic processes to determine conditions of boundedness and sample path continuity with probability one of these processes. Application of the method gives a possibility to obtain estimates for the distributions of stochastic processes. Papers by Fernique [3, 4] are among the first in this direction. In some cases, the method of majorizing measures turns out to be more effective than the entropy method exploited by Dudley [2], Fernique [4], Nanopoulos and Nobelis [14], and Kôno [5]. For example, Talagrand [15] proposed necessary and sufficient conditions in terms of majorizing measures for the sample path continuity with probability one of Gaussian stochastic processes. Such conditions in entropy terms were found by Fernique [4] for stationary Gaussian processes only. More details on the method of majorizing measures can be found in papers by Talagrand [15, 16], Ledoux and Talagrand [13], and Ledoux [12].
Particular cases of problems considered in this paper were investigated by Kozachenko and Moklyachuk [7], Kozachenko and Ryazantseva [8], Kozachenko, Vasylyk, and Yamnenko [10], Kozachenko and Sergiienko [9], Yamnenko [18]. Kozachenko and Ryazantseva [8] obtained conditions of boundedness and sample path continuity with probability one of stochastic processes from the Orlicz space of random variables generated by exponential Orlicz functions. Kozachenko, Vasylyk, and Yamnenko [10] estimated the probability that the supremum of a stochastic process from Orlicz spaces of exponential type exceeds some function. Kozachenko and Moklyachuk [7] obtained conditions of boundedness and estimates of the distribution of the supremum of stochastic processes from the Orlicz space of random variables. Kozachenko and Sergiienko [9] constructed tests for a hypothesis concerning the form of the covariance function of a Gaussian stochastic process. Yamnenko [18] obtained an estimate for distributions of norms of deviations of a stochastic process from the Orlicz space of exponential type from a given function in $L_{p}(\mathbb{T})$.
As a simple example, we apply the obtained results to a stochastic process with the same covariance function as that of the Ornstein–Uhlenbeck process but with trajectories from the $L_{2}$ space. In [17], a similar problem is considered for a generalized Ornstein–Uhlenbeck process from the Orlicz space of exponential type $\operatorname{Sub}_{\varphi }(\varOmega )$.
2 Orlicz spaces. Basic definitions
Definition 1 (Orlicz N-function [1]).
A continuous even convex function $\{U(x),x\in \mathbb{R}\}$ is said to be an Orlicz N-function if it is strictly increasing for $x>0$, $U(0)=0$, and
Any Orlicz N-function U has the following properties [11]:
Example 1.
The following functions are N-functions:
-
• $U(x)=\alpha |x{|}^{\beta },\hspace{0.2778em}\alpha >0,\hspace{0.2778em}\beta >1$;
-
• $U(x)=\exp \{|x|\}-|x|-1$;
-
• $U(x)=\exp \{\alpha |x{|}^{\beta }\}-1,\hspace{0.2778em}\alpha >0,\hspace{0.2778em}\beta >1$;
-
• $U(x)=\left\{\begin{array}{l@{\hskip10.0pt}l}{(e\alpha /2)}^{2/\alpha }{x}^{2},& |x|\le {(2/\alpha )}^{1/\alpha },\\{} \exp \{|x{|}^{\alpha }\},& |x|>{(2/\alpha )}^{1/\alpha },\hspace{1em}0<\alpha <1.\end{array}\right.$
Definition 2 (Class $\varDelta _{2}$ [11]).
Example 2.
Example 3.
-
(i) Let $U(x)=c|x{|}^{p}$, $c>0$, $p>1$. Then U belongs to the class E with constants $B=c$, $z_{0}=0$, and $D=1$.
-
(ii) The function $U(x)=|x{|}^{\beta }/{(\log (c+|x|))}^{\alpha }$ belongs to the class E if c is a number large enough such that the function $U(x)$ be convex. In this case, $z_{0}=\max \{0,\exp \{{2}^{-1/\alpha }\}-c\}$.
We will further also consider functions that belong to the intersection of the classes $\varDelta _{2}$ and E.
Example 5.
There exist functions from the class E that do not belong to the class $\varDelta _{2}$, for example, $U(x)=\exp \{|x{|}^{\alpha }\}-1$, $\alpha >1$, and $U(x)=\exp \{\phi (x)\}-1$, where $\phi (x)$ is an N-function.
Let $(\mathbb{T},\mathcal{B},\mu )$ be a measurable space with finite measure $\mu (\mathbb{T})$.
Definition 4 (Orlicz space).
The space ${L_{U}^{\mu }}(\mathbb{T})$ of measurable functions on $(\mathbb{T},\mathcal{B},\mu )$ such that, for every $f\in {L_{U}^{\mu }}(\mathbb{T})$, there exists a constant $r_{f}$ such that
is called the Orlicz space.
The space ${L_{U}^{\mu }}(\mathbb{T})$ is a Banach space with the Luxembourg norm
We will also consider the Orlicz space ${L_{U}^{\mu \times \mu }}(\mathbb{T}\times \mathbb{T})$ of measurable functions on $(\mathbb{T}\times \mathbb{T},\mathcal{B}\times \mathcal{B},\mu \times \mu )$, where $\mathcal{B}\times \mathcal{B}$ is the tensor-product sigma-algebra on the product space, and $\mu \times \mu $ is the product measure on the measurable space $(\mathbb{T}\times \mathbb{T},\mathcal{B}\times \mathcal{B})$, that is, for every $f\in {L_{U}^{\mu \times \mu }}(\mathbb{T}\times \mathbb{T})$, there exists a constant $r_{f}$ such that
Theorem 1 (Fenchel–Moreau [1]).
Let us give two examples of convex conjugate functions.
Let U be an N-function, and f be a function from the space ${L_{U}^{\mu }}(\mathbb{T})$. Consider
In the space ${L_{U}^{\mu }}(\mathbb{T})$, we can introduce a different norm equivalent to the Luxembourg norm. This is the Orlicz norm
where ${U}^{\ast }$ is the Young–Fenchel transform of the function U.
(2)
\[\| f{\| _{(U),\mu }^{\mathbb{T}}}=\underset{v:s(v;{U}^{\ast })\le 1}{\sup }\left|\int _{\mathbb{T}}f(t)\hspace{0.1667em}\mathrm{d}\mu (t)\right|,\]Lemma 1 (Hölder inequality [11]).
Lemma 2 (Krasnoselskii and Rutitskii [11]).
Let $(\varOmega ,\mathcal{F},\mathbf{P})$ be a standard probability space.
Definition 6.
The space ${L_{U}^{\mathbf{P}}}(\varOmega )=L_{U}(\varOmega )$ of random variables $\xi =\{\xi (\omega ),\omega \in \varOmega \}$ is called an Orlicz space of random variables, that is, the Orlicz space $L_{U}(\varOmega )$ is the family of random variables ξ for which that there exists a constant $r_{\xi }>0$ such that
The Luxembourg norm in this space is denoted by $\| \xi \| _{U}$, that is,
Example 7.
Suppose that $U(x)=|x{|}^{p},x\in \mathbb{R},p\ge 1$. Then $L_{U}(\varOmega )$ is the space $L_{p}(\varOmega )$, and the Luxembourg norm $\| \xi \| _{U}$ coincides with the norm $\| \xi \| _{p}={(\mathbf{E}|\xi {|}^{p})}^{1/p}$.
The following lemma follows from the Chebyshev inequality.
Lemma 3 (Buldygin and Kozachenko [1]).
3 Distribution of deviations of stochastic processes from Orlicz spaces
Let $(\mathbb{T},\rho )$ be a compact separable metric space equipped with the metric ρ, and let $\mathcal{B}$ be the Borel σ-algebra on $(\mathbb{T},\rho )$.
Consider a separable stochastic process $X=\{X(t),t\in \mathbb{T}\}$ from the Orlicz space $L_{U}(\varOmega )$, that is, $X(t)\in L_{U}(\varOmega )$, $t\in \mathbb{T}$, is continuous in the norm $\| \cdot \| _{U}$.
Note that at least one such function exists, for example,
Denote by ${\sigma }^{(-1)}(h)$ the generalized inverse to $\sigma (h)$, that is, ${\sigma }^{(-1)}(h)=\sup \{s:\sigma (s)\le h\}$. Put
and
and let S be a set from $\mathcal{B}$ such that
Consider a sequence $\epsilon _{k}(t)>0$ such that $\epsilon _{k}(t)>\epsilon _{k+1}(t)$, $\epsilon _{k}(t)\to 0$ as $k\to \infty $, and $\epsilon _{1}(t)=\sup _{s\in S}\rho (t,s)$. Put $C_{t}(u)=\{s:\rho (t,s)\le u\}$, $C_{t,k}=C_{t}(\epsilon _{k}(t))$, $\mu _{k}(t)=\mu (C_{t,k}\cap S)$.
Assumption 2.
Assume that, for a continuous function $f=\{f(t),t\in \mathbb{T}\}$, there exists a continuous increasing function $\delta (y)>0$, $y>0$, such that $\delta (y)\to 0$ as $y\to 0$ and the following condition is satisfied:
Throughout the paper, we will assume that, for all $B\in \mathcal{B}$,
Lemma 4.
Suppose that $X=\{X(t),t\in \mathbb{T}\}$ is a separable stochastic process from the Orlicz space $L_{U}(\varOmega )$ that satisfies Assumption 1. Let f be a function satisfying Assumption 2, let $\zeta (y),y>0$, be an arbitrary continuous increasing function such that $\zeta (y)\to 0$ as $y\to 0$, and let
Then, for any $S\in \mathcal{B}$ satisfying (6), we have the following inequality with probability one:
(7)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \underset{t\in S}{\sup }\left|X(t)-f(t)-\frac{1}{\mu (S)}\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|\\{} & \displaystyle \hspace{1em}\le \hspace{0.1667em}{\left\| \frac{X(u)\hspace{0.1667em}-\hspace{0.1667em}X(v)\hspace{0.1667em}-\hspace{0.1667em}f(u)\hspace{0.1667em}+\hspace{0.1667em}f(v)}{\zeta (d_{f}(u,v))}\right\| _{U,\mu \times \mu }^{S\times S}}\underset{t\in S}{\sup }\sum \limits_{l=1}^{\infty }\zeta \big(2\sigma \big(\epsilon _{l}(t)\big)\big){U}^{(-1)}\bigg(\frac{1}{{\mu _{l+1}^{2}}(t)}\bigg).\end{array}\]Proof.
Let V be the set of separability of the process X, and let t be an arbitrary point from $S\cap V$. Put
where $\chi _{A}(u)$ is the indicator function of A. Then
as $l\to \infty $. If follows from Lemma 3 and (8) that
in probability as $l\to \infty $. Therefore, there exists a sequence $l_{n}$ such that
with probability one as $l_{n}\to \infty $. It is easy to see that
It follows from (9) that the following inequality holds with probability one:
(8)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \left\| X(t)-f(t)-\int _{S}\big(X(u)-f(u)\big)\tau _{l}(u)\hspace{0.1667em}\mathrm{d}\mu (u)\right\| _{U}\\{} & \displaystyle \hspace{1em}\le \int _{S}\big\| \big(X(t)-X(u)\big)-\big(f(t)-f(u)\big)\big\| _{U}\tau _{l}(u)\hspace{0.1667em}\mathrm{d}\mu (u)\\{} & \displaystyle \hspace{1em}\le \int _{S}\big\| X(t)-X(u)\big\| _{U}\tau _{l}(u)\hspace{0.1667em}\mathrm{d}\mu (u)+\int _{S}\big|f(t)-f(u)\big|\tau _{l}(u)\hspace{0.1667em}\mathrm{d}\mu (u)\\{} & \displaystyle \hspace{1em}\le \sigma \big(\epsilon _{l}(t)\big)+\delta \big(\sigma (\epsilon _{l}(t))\big)\to 0\end{array}\](9)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \left|X(t)-f(t)-\int _{S}\big(X(u)-f(u)\big)\tau _{l}(u)\hspace{0.1667em}\mathrm{d}\mu (u)\right|\hspace{2em}\\{} & \displaystyle \hspace{1em}=\bigg|X(t)-f(t)-\int _{S}\big(X(u)-f(u)\big)\tau _{l_{n}}(u)\hspace{0.1667em}\mathrm{d}\mu (u)\\{} & \displaystyle \hspace{2em}+\sum \limits_{l=1}^{l_{n}-1}\left(\int _{S}\big(X(u)-f(u)\big)\tau _{l+1}(u)\hspace{0.1667em}\mathrm{d}\mu (u)-\int _{S}\big(X(u)-f(u)\big)\tau _{l}(u)\hspace{0.1667em}\mathrm{d}\mu (u)\right)\bigg|\\{} & \displaystyle \hspace{1em}\le \bigg|X(t)-f(t)-\int _{S}\big(X(u)-f(u)\big)\tau _{l_{n}}(u)\hspace{0.1667em}\mathrm{d}\mu (u)\bigg|\\{} & \displaystyle \hspace{2em}+\sum \limits_{l=1}^{l_{n}-1}\bigg|\int _{S}\big(X(u)-f(u)\big)\tau _{l+1}(u)\hspace{0.1667em}\mathrm{d}\mu (u)-\int _{S}\big(X(u)-f(u)\big)\tau _{l}(u)\hspace{0.1667em}\mathrm{d}\mu (u)\bigg|.\end{array}\](10)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \left|X(t)-f(t)-\frac{1}{\mu (S)}\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|\\{} & \displaystyle \hspace{1em}\le \sum \limits_{l=1}^{\infty }\bigg|\int _{S}\big(X(u)-f(u)\big)\tau _{l+1}(u)\hspace{0.1667em}\mathrm{d}\mu (u)-\int _{S}\big(X(u)-f(u)\big)\tau _{l}(u)\hspace{0.1667em}\mathrm{d}\mu (u)\bigg|\\{} & \displaystyle \hspace{1em}=\sum \limits_{l=1}^{\infty }\bigg|\int _{S}\int _{S}\big(X(u)-X(v)-f(u)+f(v)\big)\tau _{l+1}(u)\tau _{l}(v)\hspace{0.1667em}\mathrm{d}\mu (u)\hspace{0.1667em}\mathrm{d}\mu (v)\bigg|\\{} & \displaystyle \hspace{1em}\le \int _{S\times S}\bigg|\frac{X(u)-X(v)-f(u)+f(v)}{\zeta (d_{f}(u,v))}\bigg|\\{} & \displaystyle \hspace{2em}\times \Bigg(\sum \limits_{l=1}^{\infty }\tau _{l+1}(u)\tau _{l}(v)\zeta (d_{f}(u,v))\Bigg)\hspace{0.1667em}\mathrm{d}(\mu (u)\times \mu (v)).\end{array}\]From Lemma 1 and (10) we have the inequality
(11)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \left|X(t)-f(t)-\frac{1}{\mu (S)}\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|\\{} & \displaystyle \hspace{1em}\le \hspace{0.1667em}\bigg\| \frac{X(u)\hspace{0.1667em}-\hspace{0.1667em}X(v)\hspace{0.1667em}-\hspace{0.1667em}f(u)\hspace{0.1667em}+\hspace{0.1667em}f(v)}{\zeta (d_{f}(u,v))}{\bigg\| _{U,\mu \times \mu }^{S\times S}}{\left\| \sum \limits_{l=1}^{\infty }\tau _{l+1}(u)\tau _{l}(v)\zeta \big(d_{f}(u,v)\big)\right\| _{({U}^{\ast }),\mu \times \mu }^{S\times S}}.\end{array}\]Also, we have
(12)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \tau _{l+1}(u)\tau _{l}(u)\zeta \big(d_{f}(u,v)\big)& \displaystyle \le \tau _{l+1}(u)\tau _{l}(u)\zeta \big(d_{f}(u,t)+d_{f}(u,t)\big)\\{} & \displaystyle \le \tau _{l+1}(u)\tau _{l}(u)\zeta \big(\sigma \big(\epsilon _{l}(t)\big)+\sigma \big(\epsilon _{l+1}(t)\big)\big)\\{} & \displaystyle \le \tau _{l+1}(u)\tau _{l}(u)\zeta \big(2\sigma \big(\epsilon _{l}(t)\big)\big).\end{array}\]From (11) and (12) we have that with probability one the following inequality holds:
(13)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \left|X(t)-f(t)-\frac{1}{\mu (S)}\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|\\{} & \displaystyle \hspace{1em}\le \bigg\| \frac{X(u)-X(v)-f(u)+f(v)}{\zeta (d_{f}(u,v))}{\bigg\| _{U,\mu \times \mu }^{S\times S}}\\{} & \displaystyle \hspace{2em}\times \sum \limits_{l=1}^{\infty }\zeta \big(2\sigma \big(\epsilon _{l}(t)\big)\big)\big\| \tau _{l+1}(u)\tau _{l}(v){\big\| _{({U}^{\ast }),\mu \times \mu }^{S\times S}}.\end{array}\]It follows from Lemma 2 that
(14)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \big\| \tau _{l+1}(u)\tau _{l}(v){\big\| _{({U}^{\ast }),\mu \times \mu }^{S\times S}}& \displaystyle =\frac{1}{\mu _{l}(t)\mu _{l+1}(t)}{\left\| \chi _{C_{t,l}\cap S}(u)\chi _{C_{t,l+1}\cap S}(v)\right\| _{({U}^{\ast }),\mu \times \mu }^{S\times S}}\\{} & \displaystyle ={U}^{(-1)}\left(\frac{1}{\mu _{l}(t)\mu _{l+1}(t)}\right)\le {U}^{(-1)}\bigg(\frac{1}{{\mu _{l+1}^{2}}(t)}\bigg).\end{array}\]Since $t\in S\cap V$ and $S\cap V$ is a countable set, (14) holds with probability one for all $t\in S\cap V$. The process X is separable, and therefore
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \underset{t\in S}{\sup }\left|X(t)-f(t)-\frac{1}{\mu (S)}\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|\\{} & \displaystyle \hspace{1em}=\underset{t\in S\cap V}{\sup }\left|X(t)-f(t)-\frac{1}{\mu (S)}\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|\end{array}\]
with probability one. □Remark 2.
If the right side of (7) is finite, then the measure μ is called a majorizing measure on S for the process X.
Corollary 1.
Let the assumptions of Lemma 4 be satisfied. Put
with probability one, where
and
\[\zeta _{1}(t)=\zeta \big(2\sigma \big(\epsilon _{1}(t)\big)\big)=\zeta \Big(2\sigma \Big(\underset{s\in S}{\sup }\rho (t,s)\Big)\Big)\]
and
Then, for any $0<p<1$, we have the inequality
(15)
\[\underset{t\in S}{\sup }\left|X(t)-f(t)-\frac{1}{\mu (S)}\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|\le \eta _{f}C_{p}\]Proof.
Let the sequence $\epsilon _{k}(t),k\ge 1$, be defined as
Then
and
\[\mu _{l+1}(t)=\mu \big(C_{t}\big(\epsilon _{l+1}(t)\big)\cap S\big)=\nu _{t}\big(\zeta _{1}(t){p}^{l}\big).\]
Therefore, from (7) and the following inequality we obtain the assertion of the corollary:
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \hspace{-0.1667em}\sum \limits_{l=1}^{\infty }\zeta \big(2\sigma \big(\epsilon _{l}(t)\big)\big){U}^{(-1)}\hspace{-0.1667em}\bigg(\frac{1}{{\mu _{l+1}^{2}}(t)}\bigg)& \displaystyle \hspace{0.1667em}=\hspace{0.1667em}\sum \limits_{l=1}^{\infty }\zeta _{1}(t){p}^{l-1}{U}^{(-1)}\big({\big(\nu _{t}\big(\zeta _{1}(t){p}^{l}\big)\big)}^{-2}\big)\\{} & \displaystyle \hspace{0.1667em}\le \hspace{0.1667em}\sum \limits_{l\ge 1}\frac{1}{p(1-p)}{\int _{\zeta _{1}(t){p}^{l+1}}^{\zeta _{1}(t){p}^{l}}}{U}^{(-1)}\big(\nu _{t}{(u)}^{-2}\big)\hspace{0.1667em}\mathrm{d}u\\{} & \displaystyle \hspace{0.1667em}\le \hspace{0.1667em}{\int _{0}^{\zeta _{1}(t)p}}{U}^{(-1)}\big(\nu _{t}{(u)}^{-2}\big)\hspace{0.1667em}\mathrm{d}u.\end{array}\]
□Remark 3.
We will further find additional conditions on $\eta _{f}$ and $C_{p}$ from (16) and (17) such that the constant $C_{p}$ is finite and the random variable $\eta _{f}$ is finite with probability one. In this case, we get that μ is a majorizing measure on S for X. In Theorems 3 and 4, these conditions will be formulated for processes from the class $\varDelta _{2}$ and space $L_{q}(\varOmega )$.
Theorem 2.
Let assumptions of Lemma 4 be satisfied, and let the following conditions hold:
Then, for all $x>0$, we have the inequality
where $\eta _{f}$ and $C_{p}$ are defined in (16) and (17), respectively.
(19)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbf{P}\Big\{\underset{t\in S}{\sup }\big|X(t)-f(t)\big|>x\Big\}\\{} & \displaystyle \hspace{1em}\le \underset{0\le \alpha \le 1}{\inf }\underset{0<p<1}{\inf }\bigg[{\left(U\left(\alpha x\Big/\left\| \int _{S}\big(X(u)-f(u)\big)\frac{\hspace{0.1667em}\mathrm{d}\mu (u)}{\mu (S)}\right\| _{U}\right)\right)}^{-1}\\{} & \displaystyle \hspace{2em}+\mathbf{P}\bigg\{\eta _{f}>\frac{(1-\alpha )x}{C_{p}}\bigg\}\bigg],\end{array}\]Proof.
Using Fubini’s theorem and (18), we obtain that with probability one
with probability one. Since $X(u)\in L_{U}(\varOmega )$ for $u\in S$, we have $X(u)-f(u)\in L_{U}(\varOmega )$ for $u\in S$ and
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \int _{S}\int _{S}U\left(\frac{X(u)-X(v)-f(u)+f(v)}{\zeta (d_{f}(u,v))r}\right)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)\\{} & \displaystyle \hspace{1em}\le \int _{S}\int _{S}U\left(\frac{|X(u)-X(v)|+|f(u)-f(v)|}{\zeta (d_{f}(u,v))r}\right)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)<\infty ,\end{array}\]
that is, the process
with probability one belongs to the space ${L_{U}^{\mu \times \mu }}(S\times S)$. Therefore, with probability one
\[\eta _{f}={\left\| \frac{X(u)-X(v)-f(u)+f(v)}{\zeta (d_{f}(u,v))}\right\| _{U,\mu \times \mu }^{S\times S}}\]
is a finite random variable. It follows from (15) that
(20)
\[\underset{t\in S}{\sup }\big|X(t)-f(t)\big|\le \frac{1}{\mu (S)}\left|\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|+\eta _{f}C_{p}\]
\[\frac{1}{\mu (S)}\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\in L_{U}(\varOmega ).\]
Moreover,
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \left\| \frac{1}{\mu (S)}\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right\| _{U}& \displaystyle \le \frac{1}{\mu (S)}\int _{S}\big\| X(u)-f(u)\big\| _{U}\hspace{0.1667em}\mathrm{d}\mu (u)\\{} & \displaystyle \le \underset{u\in S}{\sup }\big\| X(u)-f(u)\big\| _{U}<\infty .\end{array}\]
It follows from Lemma 3 that, for any $y>0$,
It follows from (20) that, for any $0\le \alpha \le 1$ and $x>0$,
The statement of the theorem follows from (21) and (22). □
(22)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbf{P}\left\{\underset{t\in S}{\sup }\big|X(t)-f(t)\big|>x\right\}\\{} & \displaystyle \hspace{1em}\le \mathbf{P}\left\{\frac{1}{\mu (S)}\left|\int _{S}\big(X(u)-f(u)\big)\hspace{0.1667em}\mathrm{d}\mu (u)\right|>\alpha x\right\}+\mathbf{P}\big\{\eta _{f}C_{p}>(1-\alpha )x\big\}.\end{array}\]4 Distribution of deviations of stochastic processes from classes $\varDelta _{2}$ and $\varDelta _{2}\cap E$
Definition 8.
A stochastic process $X=\{X(t),t\in \mathbb{T}\}$ belongs to the class $\varDelta _{2}$ if $X\in L_{U}(\varOmega )$, where U is an Orlicz function from the class $\varDelta _{2}$.
Theorem 3.
Suppose that $X=\{X(t),t\in \mathbb{T}\}$ is a separable stochastic process from the class $\varDelta _{2}$ that satisfies Assumption 1. Let f be a function satisfying Assumption 2, where U is the Orlicz N-function from the class $\varDelta _{2}$, let $\zeta (y),y>0$, be an arbitrary continuous increasing function such that $\zeta (y)\to 0$ as $y\to 0$, and let
Then, for any $0<p<1$, the following inequality holds with probability one:
where
is a finite with probability one random variable.
\[\frac{X(u)-X(v)-f(u)+f(v)}{\zeta (d_{f}(u,v))}\in {L_{U}^{\mu \times \mu }}(\mathbb{T}\times \mathbb{T}).\]
Suppose that the following conditions are satisfied:
-
a) there exists a constant $r>0$ such thatwhere K and $x_{0}$ are introduced in Definition 2 of the class $\varDelta _{2}$ and $\gamma (u)=u/\zeta (u)$;
(23)
\[\int _{S}\int _{S}K\bigg(\frac{\gamma (d_{f}(u,v))}{r}\bigg)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)<\infty ,\] -
b)where $\zeta _{1}(t)$ and $\nu _{t}(u)$ are defined in Corollary 1.
(24)
\[\underset{t\in S}{\sup }{\int _{0}^{\zeta _{1}(t)}}{U}^{(-1)}\big({\big(\nu _{t}(u)\big)}^{-2}\big)\hspace{0.1667em}\mathrm{d}u<\infty ,\]
(25)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \underset{t\in S}{\sup }\left|X(t)-f(t)-\int _{S}\big(X(u)-f(u)\big)\frac{\hspace{0.1667em}\mathrm{d}\mu (u)}{\mu (S)}\right|\\{} & \displaystyle \hspace{1em}\le \frac{\eta _{f}}{p(1-p)}\underset{t\in S}{\sup }{\int _{0}^{\zeta _{1}(t)p}}{U}^{(-1)}\big({\big(\nu _{t}(u)\big)}^{-2}\big)\hspace{0.1667em}\mathrm{d}u,\end{array}\]Proof.
It is easy to see that the assumptions of Lemma 4 are satisfied. Consider the function $\eta _{f}$. In order to show that it is finite with probability one, it suffices to prove that the random function
belongs to the space ${L_{U}^{\mu \times \mu }}(S\times S)$ with probability one. For this, it suffices to show that there exists a number $r>0$ such that
with probability one. It follows from Fubini’s theorem that it suffices to prove that
Since $U\in \varDelta _{2}$, using Assumption 2, we have
Therefore, for all r such that inequality (23) holds, we have the relation
Inequality (26) and the statement of Theorem 3 follows from the last relation. □
(27)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbf{E}U\bigg(\frac{(X(u)-X(v)-f(u)+f(v))}{r\zeta (d_{f}(u,v))}\bigg)\\{} & \displaystyle \hspace{1em}=\mathbf{E}\chi _{\frac{|X(u)-X(v)-f(u)+f(v)|}{d_{f}(u,v)}>x_{0}}\chi _{\frac{\gamma (d_{f}(u,v))}{r}>1}\\{} & \displaystyle \hspace{2em}\times U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{d_{f}(u,v)}\bigg)K\bigg(\frac{\gamma (d_{f}(u,v))}{r}\bigg)\\{} & \displaystyle \hspace{2em}+\mathbf{E}\chi _{\frac{|X(u)-X(v)-f(u)+f(v)|}{d_{f}(u,v)}\le x_{0}}\chi _{\frac{\gamma (d_{f}(u,v))}{r}>1}U\bigg(x_{0}\frac{\gamma (d_{f}(u,v))}{r}\bigg)\\{} & \displaystyle \hspace{2em}+\chi _{\frac{\gamma (d_{f}(u,v))}{r}\le 1}\mathbf{E}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{d_{f}(u,v)}\bigg)\\{} & \displaystyle \hspace{1em}\le \mathbf{E}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{d_{f}(u,v)}\bigg)K\bigg(\frac{\gamma (d_{f}(u,v))}{r}\bigg)\\{} & \displaystyle \hspace{2em}+U(x_{0})K\bigg(\frac{\gamma (d_{f}(u,v))}{r}\bigg)\\{} & \displaystyle \hspace{2em}+\chi _{\frac{\gamma (d_{f}(u,v))}{r}\le 1}\mathbf{E}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{d_{f}(u,v)}\bigg)\\{} & \displaystyle \hspace{1em}\le \bigg(K\bigg(\frac{\gamma (d_{f}(u,v))}{r}\bigg)+\chi _{\frac{\gamma (d_{f}(u,v))}{r}\le 1}\bigg)\mathbf{E}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{d_{f}(u,v)}\bigg)\\{} & \displaystyle \hspace{2em}+U(x_{0})K\bigg(\frac{\gamma (d_{f}(u,v))}{r}\bigg)\\{} & \displaystyle \hspace{1em}\le K\bigg(\frac{\gamma (d_{f}(u,v))}{r}\bigg)\big(1+U(x_{0})\big)+\chi _{\frac{\gamma (d_{f}(u,v))}{r}\le 1}.\end{array}\](28)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbf{E}\int _{S}\int _{S}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{r\zeta (d_{f}(u,v))}\bigg)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)\\{} & \displaystyle \hspace{1em}\le \int _{S}\int _{S}\chi _{\frac{\gamma (d_{f}(u,v))}{r}\le 1}\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)\\{} & \displaystyle \hspace{2em}+\big(1+U(x_{0})\big)\int _{S}\int _{S}K\bigg(\frac{\gamma (d_{f}(u,v))}{r}\bigg)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)<\infty .\end{array}\]Corollary 2.
Proof.
It follows from (25) and Chebyshev’s inequality that
□
(29)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbf{P}\{\eta _{f}>x\}\\{} & \displaystyle \hspace{1em}=\mathbf{P}\left\{\int _{S}\int _{S}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{x\zeta (d_{f}(u,v))}\bigg)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)>1\right\}\\{} & \displaystyle \hspace{1em}\le \mathbf{E}\int _{S}\int _{S}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{x\zeta (d_{f}(u,v))}\bigg)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)\\{} & \displaystyle \hspace{1em}\le Z(x).\end{array}\]Corollary 3.
Proof.
It follows from (28), the definition of class E, and Chebyshev’s inequality that
□
(30)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbf{P}\{\eta _{f}>x\}\\{} & \displaystyle \hspace{1em}=\mathbf{P}\bigg\{\int _{S}\int _{S}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{x\zeta (d_{f}(u,v))}\bigg)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)>1\bigg\}\\{} & \displaystyle \hspace{1em}\le \mathbf{E}\int _{S}\int _{S}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{d_{f}(u,v)}\frac{\gamma (d_{f}(u,v))}{x}\bigg)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)\\{} & \displaystyle \hspace{1em}=\frac{1}{U(\frac{x}{Dr})}\mathbf{E}\int _{S}\int _{S}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{d_{f}(u,v)}\frac{\gamma (d_{f}(u,v))}{x}\bigg)\\{} & \displaystyle \hspace{2em}\times U\bigg(\frac{x}{Dr}\bigg)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)\\{} & \displaystyle \hspace{1em}\le \frac{B}{U(x/(Dr))}\\{} & \displaystyle \hspace{2em}\times \mathbf{E}\int _{S}\int _{S}U\bigg(\frac{X(u)-X(v)-f(u)+f(v)}{d_{f}(u,v)}\frac{\gamma (d_{f}(u,v))}{r}\bigg)\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)\\{} & \displaystyle \hspace{1em}\le \frac{Z(r)B}{U(x/(Dr))}.\end{array}\]Corollary 4.
Let the assumptions of Theorem 3 be satisfied. Then
a) for all $x>r$, we have the inequality
where $Z(x)$ is determined in Corollary 2, $C_{p}$ is determined in Theorem 2, and r is a constant such that condition (23) holds;
(31)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbf{P}\Big\{\underset{t\in S}{\sup }\big|X(t)-f(t)\big|>x\Big\}\\{} & \displaystyle \hspace{1em}\le \underset{0<\alpha <1}{\inf }\underset{0<p<1}{\inf }\bigg(1/U\bigg(\frac{x\alpha }{\| \int _{S}(X(u)\hspace{0.1667em}-\hspace{0.1667em}f(u))\hspace{0.1667em}\mathrm{d}\mu (u)/\mu (S)\| _{U}}\bigg)+\hspace{0.1667em}Z\bigg(\frac{x(1\hspace{0.1667em}-\hspace{0.1667em}\alpha )}{C_{p}}\bigg)\hspace{-0.1667em}\bigg),\end{array}\]
b) if $U\in \varDelta _{2}\cap E$ with $z_{0}=0$, then, for all $x>0$, we have the inequality
where B and D are the constants determined in Definition 3, r is a constant such that condition (23) holds true and $Z(x)$ is determined in Corollary 2.
(32)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbf{P}\Big\{\underset{t\in S}{\sup }\big|X(t)-f(t)\big|>x\Big\}\\{} & \displaystyle \hspace{1em}\le \underset{0<\alpha <1}{\inf }\underset{0<p<1}{\inf }\bigg(1/U\bigg(\frac{x\alpha }{\| \int _{S}(X(u)-f(u))\hspace{0.1667em}\mathrm{d}\mu (u)/\mu (S)\| _{U}}\bigg)\\{} & \displaystyle \hspace{2em}+Z(r)B/U\bigg(\frac{x(1-\alpha )}{DrC_{p}}\bigg)\bigg),\end{array}\]Theorem 4.
Suppose that $X=\{X(t),t\in \mathbb{T}\}$ is a separable stochastic process from the space $L_{q}(\varOmega )$, $q>1$, satisfying Assumption 1. Let $f\in {L_{q}^{\mu }}(S)$ be a function satisfying Assumption 2, let $\zeta (y),y>0$, be an arbitrary continuous increasing function such that $\zeta (y)\to 0$ as $y\to 0$, and let the following conditions hold:
where
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \varDelta _{q}=\int _{S}\int _{S}{\left(\gamma \big(d_{f}(u,v)\big)\right)}^{q}\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)<\infty ,\\{} & \displaystyle \underset{t\in S}{\sup }{\int _{0}^{\zeta _{1}(t)}}{\big(\nu _{t}(u)\big)}^{-2/q}\hspace{0.1667em}\mathrm{d}u<\infty ,\end{array}\]
where $\gamma (y)=y/\zeta (y)$, $\zeta _{1}(t)$ and $\nu _{t}(u)$ are defined in Corollary 1. Then, for any $0<p<1$ and $x>0$, we have the inequality
(33)
\[\mathbf{P}\Big\{\underset{t\in S}{\sup }\big|X(t)-f(t)\big|>x\Big\}\le {x}^{-q}{\big({\varGamma _{q}^{\frac{1}{q+1}}}+{\big({D_{p,q}^{q}}\varDelta _{q}\big)}^{\frac{1}{q+1}}\big)}^{q+1},\](34)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \varGamma _{q}& \displaystyle =\mathbf{E}{\left(\int _{S}\big(X(u)-f(u)\big)\frac{\hspace{0.1667em}\mathrm{d}\mu (u)}{\mu (S)}\right)}^{q},\\{} \displaystyle D_{p,q}& \displaystyle =\underset{t\in S}{\sup }\frac{1}{p(1-p)}{\int _{0}^{\zeta _{1}(t)p}}{\big(\nu _{t}(u)\big)}^{-2/q}\hspace{0.1667em}\mathrm{d}u.\end{array}\]Proof.
Consider inequality (31). In this case,
\[\left\| \int _{S}\big(X(u)-f(u)\big)\frac{\hspace{0.1667em}\mathrm{d}\mu (u)}{\mu (S)}\right\| _{U}={\varGamma _{q}^{1/q}},\]
$B=D=1$, $x_{0}=0$, $K(y)={y}^{q}$, $r>0$,
\[C_{p}=\underset{t\in S}{\sup }\frac{1}{p(1-p)}{\int _{0}^{p\zeta _{1}(t)}}{\big(\nu _{t}(u)\big)}^{-\frac{2}{q}}\hspace{0.1667em}\mathrm{d}u,\]
and $Z(r){r}^{q}\to \varDelta _{q}$ as $r\to 0$, where
\[\begin{array}{r@{\hskip0pt}l}\displaystyle Z(r){r}^{q}& \displaystyle ={r}^{q}\int _{S}\int _{S}\chi _{\frac{\gamma (d_{f}(u,v))}{r}\le 1}\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big)\\{} & \displaystyle \hspace{1em}+\int _{S}\int _{S}{\big(\gamma \big(d_{f}(u,v)\big)\big)}^{q}\hspace{0.1667em}\mathrm{d}\big(\mu (u)\times \mu (v)\big).\end{array}\]
It follows from (31) that, for any $0<p<1$,
\[\mathbf{P}\Big\{\underset{t\in S}{\sup }\big|X(t)-f(t)\big|>x\Big\}\le \underset{0\le \alpha \le 1}{\inf }\bigg(\frac{\varGamma _{q}}{{\alpha }^{q}{x}^{q}}+\frac{{C_{p}^{q}}\varDelta _{q}}{{(1-\alpha )}^{q}{x}^{q}}\bigg).\]
Inequality (33) follows from the last inequality after taking the infimum with respect to α. □5 Example of existence of majorizing measure for $L_{2}(\varOmega )$-process
In this section, we show that the Lebesgue measure is majorizing on S for some process X from the space $L_{2}(\varOmega )$.
Let $S=\mathbb{T}=[0,T]$. Assume that $\rho (u,v)=d_{f}(u,v)=|u-v|$ and let μ be the Lebesgue measure, that is, $\mu (S)=T$. Then
and
The function $\zeta (u)={u}^{\alpha }$, $\alpha >0$, satisfies the condition of Lemma 4; therefore, $\gamma (u)={u}^{1-\alpha }$ and the expressions in Theorem 4 take the following form:
and
(35)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \nu _{t}(u)& \displaystyle =\min \left\{T,t+{\sigma }^{(-1)}\left(\frac{1}{2}{u}^{1/\alpha }\right)\right\}-\max \left\{0,t-{\sigma }^{(-1)}\left(\frac{1}{2}{u}^{1/\alpha }\right)\right\},\\{} \displaystyle \zeta _{1}(t)& \displaystyle =\zeta \Big(2\sigma \Big(\underset{s\in S}{\sup }|t-s|\Big)\Big)={\big(2\sigma \big(\max \{t,T-t\}\big)\big)}^{\alpha },\end{array}\]Let $q=2$, that is, $X(t)$ is a stochastic process from $L_{2}(\varOmega )$. Assume that X is a centered process with covariance function $R_{X}(u,v)=\mathbf{E}X(u)X(v)$. Then using Fubini’s theorem, we obtain the following representation of $\varGamma _{q}$ from Theorem 4:
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \varGamma _{q}& \displaystyle =\mathbf{E}{\bigg({\int _{0}^{T}}\big(X(u)-f(u)\big)\frac{\hspace{0.1667em}\mathrm{d}u}{T}\bigg)}^{2}\\{} & \displaystyle =\frac{1}{{T}^{2}}{\int _{0}^{T}}{\int _{0}^{T}}\mathbf{E}\big(X(u)-f(u)\big)\big(X(v)-f(v)\big)\hspace{0.1667em}\mathrm{d}v\hspace{0.1667em}\mathrm{d}u\\{} & \displaystyle =\frac{1}{{T}^{2}}{\int _{0}^{T}}{\int _{0}^{T}}R_{X}(u,v)\hspace{0.1667em}\mathrm{d}u\hspace{0.1667em}\mathrm{d}v+\frac{1}{{T}^{2}}{\bigg({\int _{0}^{T}}f(v)\hspace{0.1667em}\mathrm{d}v\bigg)}^{2}.\end{array}\]
Consider the following stochastic process.
Then from Theorem 4 we can state conditions for a majorizing measure on $[0,T]$ for the process X.
Theorem 5.
Let $X=\{X(t),t\in [0,T]\}$ be a centered separable generalized Ornstein–Uhlenbeck stochastic process from the space $L_{2}(\varOmega )$ satisfying Assumption 1, and let a function f satisfy Assumption 2 with the function $\delta (t)$, $t>0$, such that
where $\alpha \in (2/\beta _{2},1/\beta _{1}+1)$ with $\beta _{1},\beta _{2}\in (0,1)$ such that $2/\beta _{2}<1/\beta _{1}+1$. Then the Lebesgue measure is majorizing on $[0,T]$ for the process X, and, for any $0<p<1$ and $x>0$, we have the inequality
where
(36)
\[{\int _{0}^{T}}{\int _{0}^{T}}{\big(\delta \big({(u-v)}^{\beta _{1}/2}\big)\big)}^{2-2\alpha }\hspace{0.1667em}\mathrm{d}u\hspace{0.1667em}\mathrm{d}v<\infty ,\](37)
\[\mathbf{P}\Big\{\underset{t\in [0,T]}{\sup }\big|X(t)-f(t)\big|>x\Big\}\le {x}^{-2}{\Big({\varGamma _{2}^{\frac{1}{3}}}+\underset{\alpha \in (0,1)}{\inf }{({D_{p,2}^{2}}\varDelta _{2})}^{\frac{1}{3}}\Big)}^{3},\]
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \varGamma _{2}& \displaystyle =\frac{2(T\tau +{e}^{-\tau T}-1)}{{\tau }^{2}{T}^{2}}+\frac{1}{{T}^{2}}{\left({\int _{0}^{T}}f(v)\hspace{0.1667em}\mathrm{d}v\right)}^{2},\\{} \displaystyle \varDelta _{2}& \displaystyle ={\int _{0}^{T}}{\int _{0}^{T}}{\big(2{\big(\tau |u-v|\big)}^{\beta _{1}}+{\big(\delta \big({\big(2\tau (u-v)\big)}^{\beta _{1}/2}\big)\big)}^{2}\big)}^{1-\alpha }\hspace{0.1667em}\mathrm{d}u\hspace{0.1667em}\mathrm{d}v,\\{} \displaystyle D_{p,2}& \displaystyle =\frac{1}{p(1-p)}\underset{t\in [0,T]}{\sup }\bigg(\frac{2{\tau ^{\prime }}{({\tau ^{\prime }}\min \{t,T-t\})}^{\alpha \beta _{2}/2-1}}{1-2/\alpha }\\{} & \displaystyle \hspace{1em}+\frac{p{2}^{3\alpha /2}{\left(\tau \max \{t,T-t\}\right)}^{\alpha \beta _{2}/2}-{({\tau ^{\prime }}\min \{t,T-t\})}^{\alpha \beta _{2}/2}}{T}\bigg),\end{array}\]
where ${\tau ^{\prime }}=\tau {2}^{3/\beta _{2}}$.
Proof.
Let us apply the inequality
It is easy to see that, for all $0\le x<1$, we have $1-\exp \{-x\}\le x\le {x}^{\beta }$. Also, $1-\exp \{-x\}\le 1\le {x}^{\beta }$ for all $x\ge 1$.
Then, using (38), we have that
\[\begin{array}{r@{\hskip0pt}l}\displaystyle d(t,s)& \displaystyle =\big\| X(t)-X(s)\big\| _{L_{2}}={\big(\mathbf{E}{\big(X(t)-X(s)\big)}^{2}\big)}^{1/2}\\{} & \displaystyle ={\big(\mathbf{E}X{(t)}^{2}+\mathbf{E}X{(s)}^{2}-2R_{X}(t,s)\big)}^{1/2}={\big(2-2\exp \big\{-\tau |t-s|\big\}\big)}^{1/2}\\{} & \displaystyle \le {2}^{1/2}{\big(\tau |t-s|\big)}^{\beta /2},\end{array}\]
that is, the function $\sigma (h)={2}^{1/2}{(\tau h)}^{\beta /2}\ge \sup _{|t-s|\le h}d(t,s)$, $h>0$, satisfies Assumption 1. Then
Also, it is easy to see that, for the centered process X,
\[d_{f}(t,s)={\big({d}^{2}(t,s)+{\big(f(t)-f(s)\big)}^{2}\big)}^{1/2}\le {\big(2{\big(\tau |t-s|\big)}^{\beta _{1}}+{\delta }^{2}\big(d(t,s)\big)\big)}^{1/2}\]
for any $\beta _{1}\in (0,1]$ and
(40)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle {\int _{0}^{T}}{\int _{0}^{T}}R_{X}(s,t)\hspace{0.1667em}\mathrm{d}s\hspace{0.1667em}\mathrm{d}t& \displaystyle ={\int _{0}^{T}}{\int _{0}^{t}}{e}^{-\tau (t-s)}\hspace{0.1667em}\mathrm{d}s\hspace{0.1667em}\mathrm{d}t+{\int _{0}^{T}}{\int _{t}^{T}}{e}^{-\tau (s-t)}\hspace{0.1667em}\mathrm{d}s\hspace{0.1667em}\mathrm{d}t\\{} & \displaystyle =\frac{1}{\tau }{\int _{0}^{T}}\big(1-{e}^{-\tau t}-{e}^{-\tau (T-t)}+1\big)\hspace{0.1667em}\mathrm{d}t\hspace{0.1667em}=\hspace{0.1667em}\frac{2(T\tau \hspace{0.1667em}+\hspace{0.1667em}{e}^{-\tau T}\hspace{0.1667em}-\hspace{0.1667em}1)}{{\tau }^{2}}.\end{array}\]From (34) it follows that
\[\varDelta _{2}={\int _{0}^{T}}{\int _{0}^{T}}{\big(2{\big(\tau |t-s|\big)}^{\beta _{1}}+{\big(\delta \big({\big(2\tau (u-v)\big)}^{1/2}\big)\big)}^{2}\big)}^{1-\alpha }\hspace{0.1667em}\mathrm{d}u\hspace{0.1667em}\mathrm{d}v<\infty \]
if $\beta _{1}(1-\alpha )+1>0$, that is, if $\alpha <1/\beta _{1}+1$. Then
Applying (39) to (35) for some $\beta _{2}\in (0,1]$, we have that
\[\nu _{t}(u)=\min \bigg\{T,t+\frac{{u}^{\frac{2}{\alpha \beta _{2}}}}{\tau {2}^{3/\beta _{2}}}\bigg\}-\max \bigg\{0,t-\frac{{u}^{\frac{2}{\alpha \beta _{2}}}}{\tau {2}^{3/\beta _{2}}}\bigg\}.\]
Put ${\tau ^{\prime }}=\tau {2}^{3/\beta _{2}}$. It is easy to see that $\nu _{t}(u)=T$ if $T<t+\frac{{u}^{\frac{2}{\alpha \beta _{2}}}}{{\tau ^{\prime }}}$ and $0>t-\frac{{u}^{\frac{2}{\alpha \beta _{2}}}}{{\tau ^{\prime }}}$, that is, if $u>{({\tau ^{\prime }}\max \{t,T-t\})}^{\alpha \beta _{2}/2}$; $\nu _{t}(u)=t+\frac{{u}^{\frac{2}{\alpha \beta _{2}}}}{{\tau ^{\prime }}}-(t-\frac{{u}^{\frac{2}{\alpha \beta _{2}}}}{{\tau ^{\prime }}})=\frac{{u}^{\frac{2}{\alpha \beta _{2}}}}{2{\tau ^{\prime }}}$ if $u\le {({\tau ^{\prime }}\min \{t,T-t\})}^{\alpha \beta _{2}/2}$; and $\nu _{t}(u)=\max \{t,T-t\}+\frac{{u}^{\frac{2}{\alpha \beta _{2}}}}{{\tau ^{\prime }}}$ if ${({\tau ^{\prime }}\min \{t,T-t\})}^{\alpha \beta _{2}/2}\le u<{({\tau ^{\prime }}\max \{t,T-t\})}^{\alpha \beta _{2}/2}$.Consider
□
\[D_{p,2}=\underset{t\in [0,T]}{\sup }\frac{1}{p(1-p)}{\int _{0}^{p{2}^{3\alpha /2}{\left(\tau \max \{t,T-t\}\right)}^{\alpha \beta _{2}/2}}}\frac{1}{\nu _{t}(u)}\hspace{0.1667em}\mathrm{d}u.\]
For $\alpha >2/\beta _{2}$, we have
