VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA64 10.15559/16-VMSTA64 Research Article Averaged deviations of Orlicz processes and majorizing measures YamnenkoRostyslavyamnenko@univ.kiev.ua Taras Shevchenko National University of Kyiv, Ukraine 2016 1111201633249268 792016 28102016 28102016 © 2016 The Author(s). Published by VTeX2016 Open access article under the CC BY license.

This paper is devoted to investigation of supremum of averaged deviations |X(t)−f(t)−∫T(X(u)−f(u))dμ(u)/μ(T)| of a stochastic process from Orlicz space of random variables using the method of majorizing measures. An estimate of distribution of supremum of deviations |X(t)−f(t)| is derived. A special case of the Lq space is considered. As an example, the obtained results are applied to stochastic processes from the L2 space with known covariance functions.

 Orlicz space Orlicz process supremum distribution method of majorizing measures Ornstein–Uhlenbeck process 60G07 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: supt∈TX(t)−f(t)−1μ(T)∫T(X(u)−f(u))dμ(u) where (T,B,μ) is a measurable space with finite measure μ(T)<∞ , 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 supt∈T|X(t)−f(t)| . A special attention is devoted to the Orlicz spaces such as the Lq 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 Lp(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 L2 space. In [17], a similar problem is considered for a generalized Ornstein–Uhlenbeck process from the Orlicz space of exponential type Subφ(Ω) .

 Orlicz spaces. Basic definitions (Orlicz <italic>N</italic>-function [<xref ref-type="bibr" rid="j_vmsta64_ref_001">1</xref>]).

A continuous even convex function {U(x),x∈R} is said to be an Orlicz N-function if it is strictly increasing for x>0 0$]]>, U(0)=0 , and U(x)x→0asx→0andU(x)x→∞asx→∞.

Any Orlicz N-function U has the following properties [11]:

U(αx)≤αU(x) for any 0≤α≤1 ;

U(x)+U(y)≤U(|x|+|y|) ;

the function U(x)/x increases for x>0 0$]]>.

The following functions are N-functions:

U(x)=α|x|β,α>0,β>1 0,\hspace{0.2778em}\beta >1$]]>;

U(x)=exp{|x|}−|x|−1 ;

U(x)=exp{α|x|β}−1,α>0,β>1 0,\hspace{0.2778em}\beta >1$]]>;

U(x)=(eα/2)2/αx2,|x|≤(2/α)1/α,exp{|x|α},|x|>(2/α)1/α,0<α<1. {(2/\alpha )}^{1/\alpha },\hspace{1em}0<\alpha <1.\end{array}\right.$]]>

(Class <inline-formula id="j_vmsta64_ineq_025"><alternatives> <mml:math><mml:msub><mml:mrow><mml:mi mathvariant="italic">Δ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> <tex-math><![CDATA[$\varDelta _{2}$]]></tex-math></alternatives></inline-formula> [<xref ref-type="bibr" rid="j_vmsta64_ref_011">11</xref>]).

An N-function U(x) belongs to the class Δ2 if there exist a constant x0≥0 and an increasing function K(x)>0 0$]]>, x≥0 , such that U(zx)≤K(z)U(x)forz≥1,x≥x0.

The following functions are from the class Δ2 :

U(x)=|x|α/α,α>1 1$]]>;

U(x)=|x|α(|ln|x||+1),α>1 1$]]>;

U(x)=(1+|x|)(|ln(1+|x|)+1)−|x| .

 The function U(x)=exp{|x|}−|x|−1 increases faster than any power function, and therefore it does not belong to the class Δ2 .

 (Class <italic>E</italic> [<xref ref-type="bibr" rid="j_vmsta64_ref_001">1</xref>, <xref ref-type="bibr" rid="j_vmsta64_ref_006">6</xref>]).

An N-function U(x) belongs to the class E if there exist constants z0≥0 , B>0 0$]]>, and D>0 0$]]> such that, for all x≥z0 and y≥z0 , U(x)U(y)≤BU(Dxy).

Let U(x)=c|x|p , c>0 0$]]>, p>1 1$]]>. Then U belongs to the class E with constants B=c , z0=0 , and D=1 .

The function U(x)=|x|β/(log(c+|x|))α belongs to the class E if c is a number large enough such that the function U(x) be convex. In this case, z0=max{0,exp{2−1/α}−c} .

We will further also consider functions that belong to the intersection of the classes Δ2 and E.

Let U(x)=|x|q , q>1 1$]]>. Then U∈Δ2∩E .

There exist functions from the class E that do not belong to the class Δ2 , for example, U(x)=exp{|x|α}−1 , α>1 1$]]>, and U(x)=exp{ϕ(x)}−1 , where ϕ(x) is an N-function.

Let (T,B,μ) be a measurable space with finite measure μ(T) .

(Orlicz space).

The space LUμ(T) of measurable functions on (T,B,μ) such that, for every f∈LUμ(T) , there exists a constant rf such that ∫TU(f(t)rf)dμ(t)<∞ is called the Orlicz space.

The space LUμ(T) is a Banach space with the Luxembourg norm ‖f‖U,μT=inf{r>0:∫TU(f(t)r)dμ(t)≤1}. 0:\int _{T}U\bigg(\frac{f(t)}{r}\bigg)\hspace{0.1667em}\mathrm{d}\mu (t)\le 1\bigg\}.\]]]>

We will also consider the Orlicz space LUμ×μ(T×T) of measurable functions on (T×T,B×B,μ×μ) , where B×B is the tensor-product sigma-algebra on the product space, and μ×μ is the product measure on the measurable space (T×T,B×B) , that is, for every f∈LUμ×μ(T×T) , there exists a constant rf such that ∫T∫TU(f(t,s)rf)d(μ(t)×μ(s))<∞.

(Young–Fenchel transform).

Let {U(x),x∈R} be an Orlicz N-function. The function {U∗(x),x∈R} for which U∗(x)=supy∈R(xy−U(y)) is called the Young–Fenchel transform of the function U.

If x>0 0$]]>, then U∗(x)=supy>0(xy−U(y)),U∗(−x)=U∗(x). 0}{\sup }\big(xy-U(y)\big),\hspace{2em}{U}^{\ast }(-x)={U}^{\ast }(x).\]]]>

(Fenchel–Moreau [<xref ref-type="bibr" rid="j_vmsta64_ref_001">1</xref>]).

Suppose that U is an N-function. Then (U∗)∗=U.

Let us give two examples of convex conjugate functions.

Suppose that p>1 1$]]> and q is the conjugate exponent of p: 1/p+1/q=1 . Let U(x)=|x|p/p . Then U∗(x)=|xq|/q .

Assume that U(x)=e|x|−|x|−1 , x∈R . Then U∗(x)=(1+|x|)(ln(1+|x|)+1)−|x|,x∈R.

Let U be an N-function, and f be a function from the space LUμ(T) . Consider s(f;U)=∫TU(f(t))dμ(t)<∞. In the space LUμ(T) , we can introduce a different norm equivalent to the Luxembourg norm. This is the Orlicz norm ‖f‖(U),μT=supv:s(v;U∗)≤1∫Tf(t)dμ(t), where U∗ is the Young–Fenchel transform of the function U.

(Hölder inequality [<xref ref-type="bibr" rid="j_vmsta64_ref_011">11</xref>]).

Let {f(t),t∈T} be a function from the space LUμ(T) endowed with the Luxembourg norm (1), and let {φ(t),t∈T} be a function from the space L(U∗)μ(T) endowed with the Orlicz norm (2). Then ∫T|f(t)φ(t)|dμ(t)≤‖f‖U,μT×‖φ‖(U∗),μT.

(Krasnoselskii and Rutitskii [<xref ref-type="bibr" rid="j_vmsta64_ref_011">11</xref>]).

Let U(x) be an N-function, let U∗(x) be the Young–Fenchel transform of U(x) , and let χA(t) be the indicator function of a set A⊂B . Then ‖χA‖(U∗),μT=μ(A)U(−1)(1μ(A)).

Let (Ω,F,P) be a standard probability space.

The space LUP(Ω)=LU(Ω) of random variables ξ={ξ(ω),ω∈Ω} is called an Orlicz space of random variables, that is, the Orlicz space LU(Ω) is the family of random variables ξ for which that there exists a constant rξ>0 0$]]> such that EU(ξrξ)<∞.

The Luxembourg norm in this space is denoted by ‖ξ‖U , that is, ‖ξ‖U=inf{r>0:EU(ξr)≤1}. 0:\mathbf{E}U\bigg(\frac{\xi }{r}\bigg)\le 1\bigg\}.\]]]>

Suppose that U(x)=|x|p,x∈R,p≥1 . Then LU(Ω) is the space Lp(Ω) , and the Luxembourg norm ‖ξ‖U coincides with the norm ‖ξ‖p=(E|ξ|p)1/p .

The following lemma follows from the Chebyshev inequality. (Buldygin and Kozachenko [<xref ref-type="bibr" rid="j_vmsta64_ref_001">1</xref>]).

Let ξ be a random variable from LU(Ω) . Then, for all x>0 0$]]>, P{|ξ|>x}≤Ux‖ξ‖U−1. x\big\}\le {\left(U\left(\frac{x}{\| \xi \| _{U}}\right)\right)}^{-1}.\]]]>

Let {X(t),t∈T} be a random process. The process X belongs to the Orlicz space LU(Ω) if all random variables X(t),t∈T , belong to the space LU(Ω) and supt∈T‖X(t)‖U<∞ .

Suppose that there exists a nonnegative function c(t),t∈T , such that P{|X(t)|≤c(t)}=1 , t∈T . Then X is an LU(Ω) -process for any Orlicz space LU(Ω) .

 Distribution of deviations of stochastic processes from Orlicz spaces

Let (T,ρ) be a compact separable metric space equipped with the metric ρ, and let B be the Borel σ-algebra on (T,ρ) .

Consider a separable stochastic process X={X(t),t∈T} from the Orlicz space LU(Ω) , that is, X(t)∈LU(Ω) , t∈T , is continuous in the norm ‖·‖U .

Consider such a function σ={σ(h),h>0} 0\}$]]>, t∈T , such that

σ(h)≥0 ,

σ(h) increases in h>0 0$]]>,

σ(h)→0 as h→0 ,

σ(h) is continuous, and

supρ(t,s)≤h‖X(t)−X(s)‖U≤σ(h) .

Note that at least one such function exists, for example, σ(h)=supρ(t,s)≤h‖X(t)−X(s)‖U.

Denote by σ(−1)(h) the generalized inverse to σ(h) , that is, σ(−1)(h)=sup{s:σ(s)≤h} . Put d(u,v)=‖X(u)−X(v)‖U and df(u,v)=‖X(u)−X(v)−f(u)+f(v)‖U and let S be a set from B such that (μ×μ){(u,v)∈S×S:ρ(u,v)≠0}>0. 0.\]]]> Consider a sequence ϵk(t)>0 0$]]> such that ϵk(t)>ϵk+1(t) \epsilon _{k+1}(t)$]]>, ϵk(t)→0 as k→∞ , and ϵ1(t)=sups∈Sρ(t,s) . Put Ct(u)={s:ρ(t,s)≤u} , Ct,k=Ct(ϵk(t)) , μk(t)=μ(Ct,k∩S) .

Assume that, for a continuous function f={f(t),t∈T} , there exists a continuous increasing function δ(y)>0 0$]]>, y>0 0$]]>, such that δ(y)→0 as y→0 and the following condition is satisfied: |f(u)−f(v)|≤δ(‖X(u)−X(v)‖U)≤d(u,v).

Throughout the paper, we will assume that, for all B∈B , ∫B|X(u)−f(u)|dμ(u)<∞.

Suppose that X={X(t),t∈T} is a separable stochastic process from the Orlicz space LU(Ω) that satisfies Assumption 1. Let f be a function satisfying Assumption 2, let ζ(y),y>0 0$]]>, be an arbitrary continuous increasing function such that ζ(y)→0 as y→0 , and let X(u)−X(v)−f(u)+f(v)ζ(df(u,v))∈LUμ×μ(T×T).

Then, for any S∈B satisfying (6), we have the following inequality with probability one: supt∈SX(t)−f(t)−1μ(S)∫S(X(u)−f(u))dμ(u)≤X(u)−X(v)−f(u)+f(v)ζ(df(u,v))U,μ×μS×Ssupt∈S ∑l=1∞ζ(2σ(ϵl(t)))U(−1)(1μl+12(t)).

Let V be the set of separability of the process X, and let t be an arbitrary point from S∩V . Put τl(u)=χCt,l∩S(u)μl(t), where χA(u) is the indicator function of A. Then X(t)−f(t)−∫S(X(u)−f(u))τl(u)dμ(u)U≤∫S‖(X(t)−X(u))−(f(t)−f(u))‖Uτl(u)dμ(u)≤∫S‖X(t)−X(u)‖Uτl(u)dμ(u)+∫S|f(t)−f(u)|τl(u)dμ(u)≤σ(ϵl(t))+δ(σ(ϵl(t)))→0 as l→∞ . If follows from Lemma 3 and (8) that ∫S(X(u)−f(u))τl(u)dμ(u)→X(t)−f(t) in probability as l→∞ . Therefore, there exists a sequence ln such that ∫S(X(u)−f(u))τln(u)dμ(u)→X(t)−f(t) with probability one as ln→∞ . It is easy to see that X(t)−f(t)−∫S(X(u)−f(u))τl(u)dμ(u)=|X(t)−f(t)−∫S(X(u)−f(u))τln(u)dμ(u)+ ∑l=1ln−1∫S(X(u)−f(u))τl+1(u)dμ(u)−∫S(X(u)−f(u))τl(u)dμ(u)|≤|X(t)−f(t)−∫S(X(u)−f(u))τln(u)dμ(u)|+ ∑l=1ln−1|∫S(X(u)−f(u))τl+1(u)dμ(u)−∫S(X(u)−f(u))τl(u)dμ(u)|. It follows from (9) that the following inequality holds with probability one: X(t)−f(t)−1μ(S)∫S(X(u)−f(u))dμ(u)≤ ∑l=1∞|∫S(X(u)−f(u))τl+1(u)dμ(u)−∫S(X(u)−f(u))τl(u)dμ(u)|= ∑l=1∞|∫S∫S(X(u)−X(v)−f(u)+f(v))τl+1(u)τl(v)dμ(u)dμ(v)|≤∫S×S|X(u)−X(v)−f(u)+f(v)ζ(df(u,v))|×( ∑l=1∞τl+1(u)τl(v)ζ(df(u,v)))d(μ(u)×μ(v)).

From Lemma 1 and (10) we have the inequality X(t)−f(t)−1μ(S)∫S(X(u)−f(u))dμ(u)≤‖X(u)−X(v)−f(u)+f(v)ζ(df(u,v))‖U,μ×μS×S ∑l=1∞τl+1(u)τl(v)ζ(df(u,v))(U∗),μ×μS×S.

Also, we have τl+1(u)τl(u)ζ(df(u,v))≤τl+1(u)τl(u)ζ(df(u,t)+df(u,t))≤τl+1(u)τl(u)ζ(σ(ϵl(t))+σ(ϵl+1(t)))≤τl+1(u)τl(u)ζ(2σ(ϵl(t))).

From (11) and (12) we have that with probability one the following inequality holds: X(t)−f(t)−1μ(S)∫S(X(u)−f(u))dμ(u)≤‖X(u)−X(v)−f(u)+f(v)ζ(df(u,v))‖U,μ×μS×S× ∑l=1∞ζ(2σ(ϵl(t)))‖τl+1(u)τl(v)‖(U∗),μ×μS×S.

It follows from Lemma 2 that ‖τl+1(u)τl(v)‖(U∗),μ×μS×S=1μl(t)μl+1(t)χCt,l∩S(u)χCt,l+1∩S(v)(U∗),μ×μS×S=U(−1)1μl(t)μl+1(t)≤U(−1)(1μl+12(t)).

Since t∈S∩V and S∩V is a countable set, (14) holds with probability one for all t∈S∩V . The process X is separable, and therefore supt∈SX(t)−f(t)−1μ(S)∫S(X(u)−f(u))dμ(u)=supt∈S∩VX(t)−f(t)−1μ(S)∫S(X(u)−f(u))dμ(u) with probability one.  □

If the right side of (7) is finite, then the measure μ is called a majorizing measure on S for the process X.

Let the assumptions of Lemma 4 be satisfied. Put ζ1(t)=ζ(2σ(ϵ1(t)))=ζ(2σ(sups∈Sρ(t,s))) and νt(u)=μ(Ct(σ(−1)(ζ(−1)(u)/2))∩S). Then, for any 0<p<1 , we have the inequality supt∈SX(t)−f(t)−1μ(S)∫S(X(u)−f(u))dμ(u)≤ηfCp with probability one, where ηf=X(u)−X(v)−f(u)+f(v)ζ(df(u,v))U,μ×μS×S and Cp=supt∈S1p(1−p)∫0pζ1(t)U(−1)((νt(u))−2)du.

Let the sequence ϵk(t),k≥1 , be defined as ϵk(t)=σ(−1)(ζ(−1)(ζ1(t)pk−1)). Then ζ(2σ(ϵl(t)))=ζ1(t)pl−1 and μl+1(t)=μ(Ct(ϵl+1(t))∩S)=νt(ζ1(t)pl). Therefore, from (7) and the following inequality we obtain the assertion of the corollary: ∑l=1∞ζ(2σ(ϵl(t)))U(−1)(1μl+12(t))= ∑l=1∞ζ1(t)pl−1U(−1)((νt(ζ1(t)pl))−2)≤∑l≥11p(1−p)∫ζ1(t)pl+1ζ1(t)plU(−1)(νt(u)−2)du≤∫0ζ1(t)pU(−1)(νt(u)−2)du.  □

We will further find additional conditions on ηf and Cp from (16) and (17) such that the constant Cp is finite and the random variable ηf 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 Δ2 and space Lq(Ω) .

Let assumptions of Lemma 4 be satisfied, and let the following conditions hold:

supt∈S∫0ζ1(t)U(−1)((νt(u))−2)du<∞,

there exists a constant r>0 0$]]> such that ∫S∫SEU|X(u)−X(v)|+|f(u)−f(v)|ζ(df(u,v))rd(μ(u)×μ(v))<∞.

Then, for all x>0 0$]]>, we have the inequality P{supt∈S|X(t)−f(t)|>x}≤inf0≤α≤1inf0<p<1[Uαx/∫S(X(u)−f(u))dμ(u)μ(S)U−1+P{ηf>(1−α)xCp}], x\Big\}\\{} & \displaystyle \hspace{1em}\le \underset{0\le \alpha \le 1}{\inf }\underset{0\frac{(1-\alpha )x}{C_{p}}\bigg\}\bigg],\end{array}\]]]> where ηf and Cp are defined in (16) and (17), respectively.

Using Fubini’s theorem and (18), we obtain that with probability one ∫S∫SUX(u)−X(v)−f(u)+f(v)ζ(df(u,v))rd(μ(u)×μ(v))≤∫S∫SU|X(u)−X(v)|+|f(u)−f(v)|ζ(df(u,v))rd(μ(u)×μ(v))<∞, that is, the process X(u)−X(v)−f(u)+f(v)ζ(df(u,v)) with probability one belongs to the space LUμ×μ(S×S) . Therefore, with probability one ηf=X(u)−X(v)−f(u)+f(v)ζ(df(u,v))U,μ×μS×S is a finite random variable. It follows from (15) that supt∈S|X(t)−f(t)|≤1μ(S)∫S(X(u)−f(u))dμ(u)+ηfCp with probability one. Since X(u)∈LU(Ω) for u∈S , we have X(u)−f(u)∈LU(Ω) for u∈S and 1μ(S)∫S(X(u)−f(u))dμ(u)∈LU(Ω). Moreover, 1μ(S)∫S(X(u)−f(u))dμ(u)U≤1μ(S)∫S‖X(u)−f(u)‖Udμ(u)≤supu∈S‖X(u)−f(u)‖U<∞.

It follows from Lemma 3 that, for any y>0 0$]]>, P∫S(X(u)−f(u))dμ(u)μ(S)>y≤1/Uy‖1μ(S)∫S(X(u)−f(u))dμ(u)‖U. y\right\}\le 1/U\left(\frac{y}{\| \frac{1}{\mu (S)}\int _{S}(X(u)-f(u))\hspace{0.1667em}\mathrm{d}\mu (u)\| _{U}}\right).\]]]>

It follows from (20) that, for any 0≤α≤1 and x>0 0$]]>, Psupt∈S|X(t)−f(t)|>x≤P1μ(S)∫S(X(u)−f(u))dμ(u)>αx+P{ηfCp>(1−α)x}. 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}\]]]> The statement of the theorem follows from (21) and (22).  □

 Distribution of deviations of stochastic processes from classes <inline-formula id="j_vmsta64_ineq_189"><alternatives> <mml:math><mml:msub><mml:mrow><mml:mi mathvariant="italic">Δ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> <tex-math><![CDATA[$\varDelta _{2}$]]></tex-math></alternatives></inline-formula> and <inline-formula id="j_vmsta64_ineq_190"><alternatives> <mml:math><mml:msub><mml:mrow><mml:mi mathvariant="italic">Δ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>∩</mml:mo><mml:mi mathvariant="italic">E</mml:mi></mml:math> <tex-math><![CDATA[$\varDelta _{2}\cap E$]]></tex-math></alternatives></inline-formula>

A stochastic process X={X(t),t∈T} belongs to the class Δ2 if X∈LU(Ω) , where U is an Orlicz function from the class Δ2 .

Suppose that X={X(t),t∈T} is a separable stochastic process from the class Δ2 that satisfies Assumption 1. Let f be a function satisfying Assumption 2, where U is the Orlicz N-function from the class Δ2 , let ζ(y),y>0 0$]]>, be an arbitrary continuous increasing function such that ζ(y)→0 as y→0 , and let X(u)−X(v)−f(u)+f(v)ζ(df(u,v))∈LUμ×μ(T×T). Suppose that the following conditions are satisfied:

there exists a constant r>0 0$]]> such that ∫S∫SK(γ(df(u,v))r)d(μ(u)×μ(v))<∞, where K and x0 are introduced in Definition 2 of the class Δ2 and γ(u)=u/ζ(u) ;

supt∈S∫0ζ1(t)U(−1)((νt(u))−2)du<∞, where ζ1(t) and νt(u) are defined in Corollary 1.

 Then, for any 0<p<1 , the following inequality holds with probability one: supt∈SX(t)−f(t)−∫S(X(u)−f(u))dμ(u)μ(S)≤ηfp(1−p)supt∈S∫0ζ1(t)pU(−1)((νt(u))−2)du, where ηf=X(u)−X(v)−f(u)+f(v)ζ(df(u,v))U,μ×μS×S is a finite with probability one random variable.

It is easy to see that the assumptions of Lemma 4 are satisfied. Consider the function ηf . In order to show that it is finite with probability one, it suffices to prove that the random function (X(u)−X(v)−f(u)+f(v))ζ(df(u,v)) belongs to the space LUμ×μ(S×S) with probability one. For this, it suffices to show that there exists a number r>0 0$]]> such that ∫S∫SU(X(u)−X(v)−f(u)+f(v)rζ(df(u,v)))d(μ(u)×μ(v))<∞ with probability one. It follows from Fubini’s theorem that it suffices to prove that ∫S∫SEU((X(u)−X(v)−f(u)+f(v))rζ(df(u,v)))d(μ(u)×μ(v))<∞.

Since U∈Δ2 , using Assumption 2, we have EU((X(u)−X(v)−f(u)+f(v))rζ(df(u,v)))=Eχ|X(u)−X(v)−f(u)+f(v)|df(u,v)>x0χγ(df(u,v))r>1×U(X(u)−X(v)−f(u)+f(v)df(u,v))K(γ(df(u,v))r)+Eχ|X(u)−X(v)−f(u)+f(v)|df(u,v)≤x0χγ(df(u,v))r>1U(x0γ(df(u,v))r)+χγ(df(u,v))r≤1EU(X(u)−X(v)−f(u)+f(v)df(u,v))≤EU(X(u)−X(v)−f(u)+f(v)df(u,v))K(γ(df(u,v))r)+U(x0)K(γ(df(u,v))r)+χγ(df(u,v))r≤1EU(X(u)−X(v)−f(u)+f(v)df(u,v))≤(K(γ(df(u,v))r)+χγ(df(u,v))r≤1)EU(X(u)−X(v)−f(u)+f(v)df(u,v))+U(x0)K(γ(df(u,v))r)≤K(γ(df(u,v))r)(1+U(x0))+χγ(df(u,v))r≤1. 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}\]]]> Therefore, for all r such that inequality (23) holds, we have the relation E∫S∫SU(X(u)−X(v)−f(u)+f(v)rζ(df(u,v)))d(μ(u)×μ(v))≤∫S∫Sχγ(df(u,v))r≤1d(μ(u)×μ(v))+(1+U(x0))∫S∫SK(γ(df(u,v))r)d(μ(u)×μ(v))<∞. Inequality (26) and the statement of Theorem 3 follows from the last relation.  □

Let the assumptions of Theorem 3 be satisfied. Let r be a number such that condition (23) holds. Then, for any x>r r$]]>, we have the inequality P{ηf>x}≤Z(x), x\}\le Z(x),\]]]> where Z(x)=∫S∫S[χγ(df(u,v))x≤1+(1+U(x0))K(γ(df(u,v))x)]d(μ(u)×μ(v)).

It follows from (25) and Chebyshev’s inequality that P{ηf>x}=P∫S∫SU(X(u)−X(v)−f(u)+f(v)xζ(df(u,v)))d(μ(u)×μ(v))>1≤E∫S∫SU(X(u)−X(v)−f(u)+f(v)xζ(df(u,v)))d(μ(u)×μ(v))≤Z(x). 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}\]]]>  □

Let the assumptions of Theorem 3 be satisfied. Let U(x)∈Δ2∩E and z0=0 in Definition 3. Then, for any x>0 0$]]>, we have the inequality P{ηf>x}≤Z(r)BU(x/Dr), x\}\le \frac{Z(r)B}{U(x/Dr)},\]]]> where B and D are the constants from Definition 3, and r is a constant such that condition (23) holds, Z(r) is defined in Corollary 2, and Z(r)≤μ2(S)+(1+U(x0))∫S∫SK(γ(df(u,v))r)d(μ(u)×μ(v))=Z1(r).

It follows from (28), the definition of class E, and Chebyshev’s inequality that P{ηf>x}=P{∫S∫SU(X(u)−X(v)−f(u)+f(v)xζ(df(u,v)))d(μ(u)×μ(v))>1}≤E∫S∫SU(X(u)−X(v)−f(u)+f(v)df(u,v)γ(df(u,v))x)d(μ(u)×μ(v))=1U(xDr)E∫S∫SU(X(u)−X(v)−f(u)+f(v)df(u,v)γ(df(u,v))x)×U(xDr)d(μ(u)×μ(v))≤BU(x/(Dr))×E∫S∫SU(X(u)−X(v)−f(u)+f(v)df(u,v)γ(df(u,v))r)d(μ(u)×μ(v))≤Z(r)BU(x/(Dr)). 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}\]]]>  □

Let the assumptions of Theorem 3 be satisfied. Then

a) for all x>r r$]]>, we have the inequality P{supt∈S|X(t)−f(t)|>x}≤inf0<α<1inf0<p<1(1/U(xα‖∫S(X(u)−f(u))dμ(u)/μ(S)‖U)+Z(x(1−α)Cp)), x\Big\}\\{} & \displaystyle \hspace{1em}\le \underset{0<\alpha <1}{\inf }\underset{0 where Z(x) is determined in Corollary 2, Cp is determined in Theorem 2, and r is a constant such that condition (23) holds;

b) if U∈Δ2∩E with z0=0 , then, for all x>0 0$]]>, we have the inequality P{supt∈S|X(t)−f(t)|>x}≤inf0<α<1inf0<p<1(1/U(xα‖∫S(X(u)−f(u))dμ(u)/μ(S)‖U)+Z(r)B/U(x(1−α)DrCp)), x\Big\}\\{} & \displaystyle \hspace{1em}\le \underset{0<\alpha <1}{\inf }\underset{0 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.

Statement a) follows from Theorem 2 and Corollary 2. Statement b) follows from Theorem 2 and Corollary 3.  □

Suppose that X={X(t),t∈T} is a separable stochastic process from the space Lq(Ω) , q>1 1$]]>, satisfying Assumption 1. Let f∈Lqμ(S) be a function satisfying Assumption 2, let ζ(y),y>0 0$]]>, be an arbitrary continuous increasing function such that ζ(y)→0 as y→0 , and let the following conditions hold: Δq=∫S∫Sγ(df(u,v))qd(μ(u)×μ(v))<∞,supt∈S∫0ζ1(t)(νt(u))−2/qdu<∞, where γ(y)=y/ζ(y) , ζ1(t) and νt(u) are defined in Corollary 1. Then, for any 0<p<1 and x>0 0$]]>, we have the inequality P{supt∈S|X(t)−f(t)|>x}≤x−q(Γq1q+1+(Dp,qqΔq)1q+1)q+1, 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},\]]]> where Γq=E∫S(X(u)−f(u))dμ(u)μ(S)q,Dp,q=supt∈S1p(1−p)∫0ζ1(t)p(νt(u))−2/qdu.

Consider inequality (31). In this case, ∫S(X(u)−f(u))dμ(u)μ(S)U=Γq1/q, B=D=1 , x0=0 , K(y)=yq , r>0 0$]]>, Cp=supt∈S1p(1−p)∫0pζ1(t)(νt(u))−2qdu, and Z(r)rq→Δq as r→0 , where Z(r)rq=rq∫S∫Sχγ(df(u,v))r≤1d(μ(u)×μ(v))+∫S∫S(γ(df(u,v)))qd(μ(u)×μ(v)). It follows from (31) that, for any 0<p<1 , P{supt∈S|X(t)−f(t)|>x}≤inf0≤α≤1(Γqαqxq+CpqΔq(1−α)qxq). 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.  □

 Example of existence of majorizing measure for <inline-formula id="j_vmsta64_ineq_243"><alternatives> <mml:math><mml:msub><mml:mrow><mml:mi mathvariant="italic">L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo mathvariant="normal" fence="true" stretchy="false">(</mml:mo><mml:mi mathvariant="italic">Ω</mml:mi><mml:mo mathvariant="normal" fence="true" stretchy="false">)</mml:mo></mml:math> <tex-math><![CDATA[$L_{2}(\varOmega )$]]></tex-math></alternatives></inline-formula>-process

In this section, we show that the Lebesgue measure is majorizing on S for some process X from the space L2(Ω) .

Let S=T=[0,T] . Assume that ρ(u,v)=df(u,v)=|u−v| and let μ be the Lebesgue measure, that is, μ(S)=T . Then Ct(u)={s:|t−s|≤u}=[t−u,t+u] and Ct∩S=min{T,t+u}−max{0,t−u}. The function ζ(u)=uα , α>0 0$]]>, satisfies the condition of Lemma 4; therefore, γ(u)=u1−α and the expressions in Theorem 4 take the following form: νt(u)=minT,t+σ(−1)12u1/α−max0,t−σ(−1)12u1/α,ζ1(t)=ζ(2σ(sups∈S|t−s|))=(2σ(max{t,T−t}))α, and Δq=∫0T∫0T(df(u,v))(1−α)qdudv.

Let q=2 , that is, X(t) is a stochastic process from L2(Ω) . Assume that X is a centered process with covariance function RX(u,v)=EX(u)X(v) . Then using Fubini’s theorem, we obtain the following representation of Γq from Theorem 4: Γq=E(∫0T(X(u)−f(u))duT)2=1T2∫0T∫0TE(X(u)−f(u))(X(v)−f(v))dvdu=1T2∫0T∫0TRX(u,v)dudv+1T2(∫0Tf(v)dv)2.

Consider the following stochastic process.

A stochastic process X={X(t),t∈T} is called the generalized Ornstein–Uhlenbeck process from the space L2(Ω) if X is an L2(Ω) -process with the covariance function RX(t,s)=e−τ|t−s|,τ>0. 0.\]]]>

Then from Theorem 4 we can state conditions for a majorizing measure on [0,T] for the process X.

Let X={X(t),t∈[0,T]} be a centered separable generalized Ornstein–Uhlenbeck stochastic process from the space L2(Ω) satisfying Assumption 1, and let a function f satisfy Assumption 2 with the function δ(t) , t>0 0$]]>, such that ∫0T∫0T(δ((u−v)β1/2))2−2αdudv<∞, where α∈(2/β2,1/β1+1) with β1,β2∈(0,1) such that 2/β2<1/β1+1 . Then the Lebesgue measure is majorizing on [0,T] for the process X, and, for any 0<p<1 and x>0 0$]]>, we have the inequality P{supt∈[0,T]|X(t)−f(t)|>x}≤x−2(Γ213+infα∈(0,1)(Dp,22Δ2)13)3, 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},\]]]> where Γ2=2(Tτ+e−τT−1)τ2T2+1T2∫0Tf(v)dv2,Δ2=∫0T∫0T(2(τ|u−v|)β1+(δ((2τ(u−v))β1/2))2)1−αdudv,Dp,2=1p(1−p)supt∈[0,T](2τ′(τ′min{t,T−t})αβ2/2−11−2/α+p23α/2τmax{t,T−t}αβ2/2−(τ′min{t,T−t})αβ2/2T), where τ′=τ23/β2 .

Let us apply the inequality 1−exp{−x}≤xβ,0<β≤1,x≥0. It is easy to see that, for all 0≤x<1 , we have 1−exp{−x}≤x≤xβ . Also, 1−exp{−x}≤1≤xβ for all x≥1 .

Then, using (38), we have that d(t,s)=‖X(t)−X(s)‖L2=(E(X(t)−X(s))2)1/2=(EX(t)2+EX(s)2−2RX(t,s))1/2=(2−2exp{−τ|t−s|})1/2≤21/2(τ|t−s|)β/2, that is, the function σ(h)=21/2(τh)β/2≥sup|t−s|≤hd(t,s) , h>0 0$]]>, satisfies Assumption 1. Then σ(−1)(h)=h2/β21/βτ,h>0. 0.\]]]>

Also, it is easy to see that, for the centered process X, df(t,s)=(d2(t,s)+(f(t)−f(s))2)1/2≤(2(τ|t−s|)β1+δ2(d(t,s)))1/2 for any β1∈(0,1] and ∫0T∫0TRX(s,t)dsdt=∫0T∫0te−τ(t−s)dsdt+∫0T∫tTe−τ(s−t)dsdt=1τ∫0T(1−e−τt−e−τ(T−t)+1)dt=2(Tτ+e−τT−1)τ2.

From (34) it follows that Δ2=∫0T∫0T(2(τ|t−s|)β1+(δ((2τ(u−v))1/2))2)1−αdudv<∞ if β1(1−α)+1>0 0$]]>, that is, if α<1/β1+1 . Then ∫0T∫0T((δ(u−v))β1/2)2−2αdudv<∞.

Applying (39) to (35) for some β2∈(0,1] , we have that νt(u)=min{T,t+u2αβ2τ23/β2}−max{0,t−u2αβ2τ23/β2}. Put τ′=τ23/β2 . It is easy to see that νt(u)=T if T<t+u2αβ2τ′ and 0>t−u2αβ2τ′ t-\frac{{u}^{\frac{2}{\alpha \beta _{2}}}}{{\tau ^{\prime }}}$]]>, that is, if u>(τ′max{t,T−t})αβ2/2 {({\tau ^{\prime }}\max \{t,T-t\})}^{\alpha \beta _{2}/2}$]]>; νt(u)=t+u2αβ2τ′−(t−u2αβ2τ′)=u2αβ22τ′ if u≤(τ′min{t,T−t})αβ2/2 ; and νt(u)=max{t,T−t}+u2αβ2τ′ if (τ′min{t,T−t})αβ2/2≤u<(τ′max{t,T−t})αβ2/2 .

Consider Dp,2=supt∈[0,T]1p(1−p)∫0p23α/2τmax{t,T−t}αβ2/21νt(u)du. For α>2/β2 2/\beta _{2}$]]>, we have ∫0p23α/2τmax{t,T−t}αβ2/2duνt(u)=∫0(4τmin{t,T−t})α/22τ′u−2/(αβ2)du+∫(τ′min{t,T−t})αβ2/2(τ′max{t,T−t})αβ2/2dumax{t,T−t}+u2/(αβ2)τ′+∫(τ′max{t,T−t})αβ2/2p23α/2τmax{t,T−t}αβ2/21Tdu≤2τ′1−2/(αβ2)(τ′min{t,T−t})αβ2/2−1+(τ′max{t,T−t})αβ2/2−(τ′min{t,T−t})αβ2/2max{t,T−t}+min{t,T−t}+p23α/2τmax{t,T−t}αβ2/2−(τ′max{t,T−t})αβ2/2T=2τ′(τ′min{t,T−t})αβ2/2−11−2/α+p23α/2τmax{t,T−t}αβ2/2−(τ′min{t,T−t})αβ2/2T.  □

 Acknowledgments

The author gratefully thanks Prof. Dr. Yu. Kozachenko, who provided insight and expertise for the research.

 References Buldygin, V.V., Kozachenko, Y.V.: Metric Characterization of Random Variables and Random Processes. AMS, Providence, RI (2000). MR1743716 Dudley, R.M.: Sample functions of the Gaussian process. Ann. Probab. 1, 66–103 (1973). MR0346884 Fernique, X.: Régularité de processus gaussiens. Invent. Math. 12, 304–320 (1971). MR0286166 Fernique, X.: Régularité des trajectoires des fonctions aléatoires gaussiennes. In: Ecole d’Eté de Probabilités de Saint-Flour. IV, 1974. Lect. Notes Math. vol. 480, pp. 1–96 (1971). MR0413238 Kôno, N.: Sample path properties of stochastic processes. J. Math. Kyoto Univ. 20, 295–313 (1980). MR0582169 Kozachenko, Y.: Random processes in Orlicz spaces. I. Theory Probab. Math. Stat. 30, 103–117 (1985). MR0800835 Kozachenko, Y., Moklyachuk, O.: Large deviation probabilities in terms of majorizing measures. Random Oper. Stoch. Equ. 11(1), 1–20 (2003). MR1969189. doi:10.1163/156939703322003953 Kozachenko, Y., Ryazantseva, V.: Conditions for boundedness and continuity in terms of majorizing measures of random processes in certain Orlicz space. Theory Probab. Math. Stat. 44, 77–83 (1992). MR1168430 Kozachenko, Y., Sergiienko, M.: The criterion of hypothesis testing on the covariance function of a Gaussian stochastic process. Monte Carlo Methods Appl. 20(1), 137–145 (2014). MR3213591. doi:10.1515/mcma-2013-0023 Kozachenko, Y., Vasylyk, O., Yamnenko, R.: Upper estimate of overrunning by Subφ(Ω) random process the level specified by continuous function. Random Oper. Stoch. Equ. 13(2), 111–128 (2005). MR2152102. doi:10.1163/156939705323383832 Krasnoselskii, M.A., Rutitskii, Y.B.: Convex Functions and Orlicz Spaces. P. Noordhoff Ltd., Groningen (1961). MR0126722 Ledoux, M.: Isoperimetry and Gaussian analysis. In: Lectures on probability theory and statistics, Ecole d’Eté de Probabilités de Saint-Flour, Lect. Notes Math., vol. 1648, pp. 165–294 (1996). MR1600888. doi:10.1007/BFb0095676 Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, Berlin, New York (1991). MR1102015. doi:10.1007/978-3-642-20212-4 Nanopoulos, C., Nobelis, P.: Régularité et propriétés limites des fonctions aléatoires. Lect. Notes Math. 649, 567–690 (1978). MR0520031 Talagrand, M.: Regularity of Gaussian processes. Acta Math. 159, 99–149 (1987). MR0906527. doi:10.1007/BF02392556 Talagrand, M.: Majorizing measures: The generic chaining. Ann. Probab. 24, 1049–1103 (1996). MR1411488. doi:10.1214/aop/1065725175 Yamnenko, R.: An estimate of the probability that the queue length exceeds the maximum for a queue that is a generalized Ornstein–Uhlenbeck stochastic process. Theory Probab. Math. Stat. 73, 181–194 (2006). MR2213851. doi:10.1090/S0094-9000-07-00691-6 Yamnenko, R.: A bound for norms in Lp(T) of deviations of φ-sub-Gaussian stochastic processes. Lith. Math. J. 55(2), 291–300 (2015). MR3347597. doi:10.1007/s10986-015-9281-0