Averaged deviations of Orlicz processes and majorizing measures

This paper is devoted to investigation of supremum of averaged deviations $|X(t)-f(t)-\int_{\mathbb {T}}(X(u)-f(u))\,\mathrm {d}\mu(u)/\mu(\mathbb {T})|$ 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 $L_q$ space is considered. As an example, the obtained results are applied to stochastic processes from the $L_2$ space with known covariance functions.


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: 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 sup t∈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 (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 Sub ϕ (Ω).
We will further also consider functions that belong to the intersection of the classes ∆ 2 and E.

Example 5.
There exist functions from the class E that do not belong to the class ∆ 2 , for example, Let (T, B, µ) be a measurable space with finite measure µ(T).
Definition 4 (Orlicz space). The space L µ U (T) of measurable functions on (T, B, µ) such that, for every f ∈ L µ U (T), there exists a constant r f such that We will also consider the Orlicz space L µ×µ U (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 ∈ L µ×µ U (T × T), there exists a constant r f such that is called the Young-Fenchel transform of the function U .
Theorem 1 (Fenchel-Moreau [1]). Suppose that U is an N-function. Then Let us give two examples of convex conjugate functions.
Let U be an N -function, and f be a function from the space L µ U (T). Consider In the space L µ U (T), we can introduce a different norm equivalent to the Luxembourg norm. This is the Orlicz norm where U * is the Young-Fenchel transform of the function U .
Lemma 1 (Hölder inequality [11]). Let {f (t), t ∈ T} be a function from the space L µ U (T) endowed with the Luxembourg norm (1), and let {ϕ(t), t ∈ T} be a function from the space L µ (U * ) (T) endowed with the Orlicz norm (2). Then Lemma 2 (Krasnoselskii and Rutitskii [11]). Let U (x) be an N-function, let U * (x) be the Young-Fenchel transform of U (x), and let χ A (t) be the indicator function of a set A ⊂ B. Then Let (Ω, F , P) be a standard probability space.
, ω ∈ Ω} is called an Orlicz space of random variables, that is, the Orlicz space L U (Ω) is the family of random variables ξ for which that there exists a constant r ξ > 0 such that The Luxembourg norm in this space is denoted by ξ U , that is, , 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.

Lemma 3 (Buldygin and Kozachenko [1]). Let ξ be a random variable from L U (Ω).
Then, for all x > 0, Definition 7. Let {X(t), t ∈ T} be a random process. The process X belongs to the Orlicz space L U (Ω) if all random variables X(t), t ∈ T, belong to the space L U (Ω) and sup t∈T X(t) U < ∞.
Then X is an L U (Ω)-process for any Orlicz space L U (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 L U (Ω), that is, X(t) ∈ L U (Ω), t ∈ T, is continuous in the norm · U .
Note that at least one such function exists, for example, and let S be a set from B such that Assumption 2. Assume that, for a continuous function f = {f (t), t ∈ T}, there exists a continuous increasing function δ(y) > 0, y > 0, such that δ(y) → 0 as y → 0 and the following condition is satisfied: Throughout the paper, we will assume that, for all B ∈ B,

Lemma 4.
Suppose that X = {X(t), t ∈ T} is a separable stochastic process from the Orlicz space L U (Ω) that satisfies Assumption 1. Let f be a function satisfying Assumption 2, let ζ(y), y > 0, be an arbitrary continuous increasing function such that ζ(y) → 0 as y → 0, and let Then, for any S ∈ B satisfying (6), we have the following inequality with probability one: Proof. Let V be the set of separability of the process X, and let t be an arbitrary point from S ∩ V . Put where χ A (u) is the indicator function of A. Then as l → ∞. If follows from Lemma 3 and (8) that in probability as l → ∞. Therefore, there exists a sequence l n such that with probability one as l n → ∞. It is easy to see that It follows from (9) that the following inequality holds with probability one: From Lemma 1 and (10) we have the inequality Also, we have From (11) and (12) we have that with probability one the following inequality holds: It follows from Lemma 2 that 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 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.
Therefore, from (7) and the following inequality we obtain the assertion of the corollary: Remark 3. We will further find additional conditions on η f and C p from (16) and (17) such that the constant C p is finite and the random variable η 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 ∆ 2 and space L q (Ω).

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 η f and C p are defined in (16) and (17), respectively.
Proof. Using Fubini's theorem and (18), we obtain that with probability one with probability one belongs to the space L µ×µ U (S × S). Therefore, with probability one is a finite random variable. It follows from (15) that Moreover, It follows from Lemma 3 that, for any y > 0, It follows from (20) that, for any 0 ≤ α ≤ 1 and x > 0, The statement of the theorem follows from (21) and (22).

Distribution of deviations of stochastic processes from classes ∆ 2 and ∆ 2 ∩E
where U is an Orlicz function from the class ∆ 2 .

Theorem 3.
Suppose that X = {X(t), t ∈ T} is a separable stochastic process from the class ∆ 2 that satisfies Assumption 1. Let f be a function satisfying Assumption 2, where U is the Orlicz N-function from the class ∆ 2 , let ζ(y), y > 0, be an arbitrary continuous increasing function such that ζ(y) → 0 as y → 0, and let Suppose that the following conditions are satisfied: a) there exists a constant r > 0 such that where K and x 0 are introduced in Definition 2 of the class ∆ 2 and γ(u) where ζ 1 (t) and ν t (u) are defined in Corollary 1.
Then, for any 0 < p < 1, the following inequality holds with probability one: where is a finite with probability one random variable.
Proof. It is easy to see that the assumptions of Lemma 4 are satisfied. Consider the function η 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 µ×µ U (S × 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 ∈ ∆ 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.

Corollary 2.
Let the assumptions of Theorem 3 be satisfied. Let r be a number such that condition (23) holds. Then, for any x > r, we have the inequality

Proof. It follows from (25) and Chebyshev's inequality that
Corollary 3. Let the assumptions of Theorem 3 be satisfied. Let U (x) ∈ ∆ 2 ∩ E and z 0 = 0 in Definition 3. Then, for any x > 0, we have the inequality 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 Proof. It follows from (28), the definition of class E, and Chebyshev's inequality that . (30)

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; b) if U ∈ ∆ 2 ∩ 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.
Proof. Statement a) follows from Theorem 2 and Corollary 2. Statement b) follows from Theorem 2 and Corollary 3.

Theorem 4.
Suppose that X = {X(t), t ∈ T} is a separable stochastic process from the space L q (Ω), q > 1, satisfying Assumption 1. Let f ∈ L µ q (S) be a function satisfying Assumption 2, let ζ(y), y > 0, be an arbitrary continuous increasing function such that ζ(y) → 0 as y → 0, and let the following conditions hold: where γ(y) = y/ζ(y), ζ 1 (t) and ν t (u) are defined in Corollary 1. Then, for any 0 < p < 1 and x > 0, we have the inequality where Proof. Consider inequality (31). In this case, and Z(r)r q → ∆ q as r → 0, where It follows from (31) that, for any 0 < p < 1, Inequality (33) follows from the last inequality after taking the infimum with respect to α.

Example of existence of majorizing measure for L 2 (Ω)-process
In this section, we show that the Lebesgue measure is majorizing on S for some process X from the space L 2 (Ω).
Let S = T = [0, T ]. Assume that ρ(u, v) = d f (u, v) = |u − v| and let µ be the Lebesgue measure, that is, µ(S) = T . Then The function ζ(u) = u α , α > 0, satisfies the condition of Lemma 4; therefore, γ(u) = u 1−α and the expressions in Theorem 4 take the following form: Let q = 2, that is, X(t) is a stochastic process from L 2 (Ω). Assume that X is a centered process with covariance function R X (u, v) = EX(u)X(v). Then using Fubini's theorem, we obtain the following representation of Γ q from Theorem 4: Consider the following stochastic process.

Definition 9.
A stochastic process X = {X(t), t ∈ T} is called the generalized Ornstein-Uhlenbeck process from the space L 2 (Ω) if X is an L 2 (Ω)-process with the covariance function R X (t, s) = e −τ |t−s| , τ > 0.
Then from Theorem 4 we can state conditions for a majorizing measure on [0, T ] for the process X.