Abstract

We introduce the notion of almost automorphic random functions in probability. Some basic and fundamental properties of such functions are established.

1. Introduction

Since almost periodic random functions in probability are introduced in [1], several authors have made contributions on such almost periodic random functions (see, e.g., [2–5] and references therein). In fact, almost periodic random functions in probability are a natural generalization of the deterministic almost periodic functions.

On the other hand, just like that almost periodic functions are an important generalization of continuous periodic functions (see, e.g., [6], where it is shown that the space of continuous periodic functions is a set of first category in ), almost automorphic functions are an important generalization of almost periodic functions (cf. [7, 8] for some basic results and applications about almost automorphic functions). However, to the best of our knowledge, the notion of almost automorphic random functions in probability has not been introduced and studied. So, in this paper, we aim to study some basic and fundamental properties of almost automorphic random functions in probability.

It is needed to note that another kind of almost periodic random functions, which is called th mean almost periodic random functions, has been introduced and studied by Bezandry and Diagana. We refer the reader to the monograph of Bezandry and Diagana [9] for a detailed knowledge on th mean almost periodic random functions. In addition, the notion of square-mean almost automorphic random functions is introduced recently by Fu and Liu [10]; the notion of square-mean pseudo almost automorphic random functions is introduced by Chen and Lin [11], and the concept of distributional almost automorphy for stochastic processes is introduced by Fu and Chen in [12]. For other results concerning such functions and their applications, we refer the reader to [13–15] and references therein for some recent works. But, as one will see in the next section, our notion of almost automorphic random functions in probability is different from the notion of square-mean almost automorphic random functions.

2. Almost Automorphic Random Functions in Probability

Throughout the rest of this paper, let be a probability space, the set of real numbers, and the set of positive integers.

Definition 1. A random function is called almost automorphic in probability provided that, for every sequence of real numbers , there exist a subsequence and a random function , such that, for every , , and , there corresponds a positive integer with the property that, for all , We denote all such functions by .

Remark 2. In the above definition, we do not assume that is continuous in probability on since our main interest here is the recurrent property of such functions.

Before we make further study on the properties of almost automorphic random functions in probability, we would like first to compare our notion with the notion of square-mean almost automorphism.

Definition 3 (see [10]). A random function is called square-mean almost automorphic provided that , where and means the space of all classical almost automorphic functions from to (cf. [7]). For convenience, we denote the set of all such functions by .

Remark 4. It is not difficult to show that if and if is continuous in probability on , then by using the definitions. However, the contrary is not true in general. In fact, for a random function from to , convergence in probability does not necessarily mean square-mean convergence.

Definition 5. A random function is said to be bounded in probability, if for every there exists a number such that

Theorem 6. Let . Then, the following assertions hold true:(i);(ii)for every , ;(iii)for every , ;(iv) is bounded in probability, and is also bounded in probability, where is the function in Definition 1;(v);(vi) provided that is bounded in probability;(vii)if there exists a constant such that for every

Proof. One can show (i)–(iii) by the definition of . We omit the details here. Next, let us prove (iv). We prove it by contradiction. Suppose that is not bounded in probability. Then, there exists a number such that for every , there corresponds a number with the property that Since , there exist a subsequence of (for convenience, we still denote it by ) and a random function , such that for the above , there corresponds a positive integer with the property that On the other hand, it is easy to see that there exists a positive integer such that Combining the above two inequalities, we get for all which contradicts with (5). In addition, the boundedness of follows from the boundedness of and the fact that, for every and , there exists such that
Now, let us prove (v). By the definition, we know that for every sequence of real numbers , there exist a subsequence and two random functions , such that, for every , , and , there corresponds a positive integer with the property that for all By (iv), we know that for every there exists a number such that for all . Then, for every , , , and , we have and by a similar argument, we can also get Thus, .
The proof of (vi) is similar to that of (v). So we omit the details.
It remains to show (vii). By Definition 1, we can choose a subsequence with and a random function , such that for every , , and there corresponds a positive integer with the property that, for all , Noting that for sufficiently large and we conclude by (14) that, for every , , and , there holds which means that for every . Combining this with (15), by a similar argument to the above proof, we can get for all .

Theorem 7. Let be uniformly convergent in probability on to a random function . Then, .

Proof. For every sequence of real numbers , by the diagonal method, one can choose a subsequence such that, for every , there is a random function , that satisfying for every , , and , there corresponds a positive integer with the property that, for all , By using the fact that is uniformly convergent in probability on to , (20), and we can prove that, for every and , there exists a positive integer such that, for all and , there holds Then, by [9, Proposition 3.7], there exists a subsequence (we still denote it by for convenience) such that for every , exists for a.e. .
Let . Then, again by the fact that is uniformly convergent in probability on to , (20), and we conclude that, for every , , and , there corresponds a positive integer with the property that, for all , Similarly, we can obtain Thus, . This completes the proof.

Lemma 8. Let . Then, for every and , there exist finite real numbers such that where

Proof. We prove it by contradiction. Assume that there do not exist finite real numbers such that Taking an arbitrary real number , there exists a real number ; that is, . Then, we can choose a real number ; that is, Continuing by this way, we can get a real number sequence satisfying that for every with , there holds Now, since , there exist and a random function such that Again by , one can choose a positive integer such that Combining the above two inequalities, we get Then, we have , which contradicts with (31). This completes the proof.

Theorem 9. Let and let be the set as in Lemma 8. Then, for every , there exists such that for every there holds

Proof. By Lemma 8, there exist finite real numbers such that Fix . Let and for and , It is easy to see that for every and , . So, for every , we have

Definition 10. Let . A random function is called almost automorphic in probability uniformly for provided that for every sequence of real numbers , there exist a subsequence and a random function , such that for every , , , and there corresponds a positive integer with the property that, for all , We denote all such functions by . In addition, we denote by all the random functions in the definition.

Next, for convenience, we denote by the set of all random functions satisfying that there exist a constant and with such that for all , , and .

Lemma 11. Let . Then for every and there exists finite numbers such that

Proof. Since
there exists such that . On the other hand, there exist finite numbers such that . Noting that we conclude that

Theorem 12. Let with  , and let . Then, , where for and .

Proof. Let be a sequence of real numbers. Then, there exist a subsequence and two random functions and . Here, is the function in Definition 10 and corresponds in Definition 1 ( corresponds in Definition 1).
Fix and . By Lemma 11, there exist finite numbers such that Since , there exist a constant and with such that for all , , and . In addition, there exists such that, for all , there hold Let Then we have and . Combining all the above assertions, for all , we have Similarly, one can show that for sufficiently large . Thus, we conclude that .