On the oscillation of solutions for a class of second-order nonlinear stochastic difference equations
© Yu et al.; licensee Springer. 2014
Received: 30 November 2013
Accepted: 2 March 2014
Published: 18 March 2014
In this paper, we investigate the asymptotical behavior for a partial sum sequence of independent random variables, and we derive a law of the iterated logarithm type. It is worth to point out that the partial sum sequence needs not to be an independent increment process. As an application of the theory established, we also give a sufficient criterion on the almost sure oscillation of solutions for a class of second-order stochastic difference equation of neutral type.
where is determined by and . The proof of (1.6) is to be given in Section 4. Denoting for any , it implies . It is obvious that is not independent of even though is a sequence of independent random variables. That is to say that does not have the independent increment property, which means that we do not directly adopt law of the iterated logarithm to to obtain their limit behavior. However, under some restrictions we can use a roundabout way to analyze the limit behavior on by the law of the iterated logarithm, then we give some sufficient conditions on the almost sure oscillation for (1.3). These results and proofs are deferred to the following sections.
2 Definitions and assumptions
Throughout this paper, the following notation, definitions, and assumptions are needed. ℕ and ℝ denote, respectively, the positive integer numbers and real numbers. Let for every . denotes a complete probability space. is a random variable sequence defined on . We suppose that filtration is naturally generated, namely that . We use the standard abbreviations ‘a.s.’ and ‘i.i.d.’ instead of ‘almost surely’ and ‘independent identically distribution’, respectively. For simplicity, we denote throughout this paper.
(A2.1) for every ,
(A2.2) for every ,
(A2.3) F is assumed to be Borel measurable and to obey for , and ,
(A2.4) is assumed to be independent identically distributed random variable sequence defined on , and, moreover, , .
Definition 2.1 is called a solution of (1.3) with initial values , . is constituted by and , , where is obtained by steps of iteration of (1.3) with initial values , .
where ‘i.o.’ stands for infinitely often.
Definition 2.3 Equation (1.3) is said to be a.s. oscillatory if its any solution is a.s. oscillation.
3 Law of the iterated logarithm
The classical Kolmogorov law of the iterated logarithm is an effective tool in studying the limit behavior of partial sum of independent random variable sequence (see ). In 1973, Chow and Teicher  generalized the classical results and obtained the following law of the iterated logarithm for weighted averages.
Theorem 3.1 (Iterated logarithm laws of weighted averages)
On the above results, notice that is independent of .
Note that is a monotony increasing sequence, hence we give the following hypothesis:
(C.1) There exist constants and such that .
Setting in the above inequalities, we obtain (3.3).
Taking the limit on both sides of the above equation, result (3.4) follows.
Equation (3.5) can be proved similarly. □
Lemma 3.3 (Law of the iterated logarithm on defined by (3.1))
Here α is described as (C.1).
Hence (3.11) holds.
Equation (3.12) can be proved similarly. □
To proceed the study, we give another assumption:
(C.2) There exist such that .
Note that in (C.2). It is obvious that the conclusions of Lemma 3.2 and Lemma 3.3 are also right when (C.2) replaces (C.1).
4 The main results
In this section, we give the main results on the oscillation of the solution of (1.3).
Theorem 4.1 Suppose that (1.3) satisfies, respectively, (A2.1)-(A2.4), then (1.3) is an almost sure oscillation under condition (C.2).
Proof Suppose that the result is not right, then (1.3) must have a solution, denoted as , and it is not an almost sure oscillation. That is to say at least one is not true between and .
Therefore the left-hand side of (4.4) is nonnegative.
due to (). Therefore we find that it is an oscillation by (4.3). This is a contradiction.
2. Secondly, we assume that holds. We may get a contradiction for case 2 similar to case 1. Thus we finish the proof of Theorem 4.1. □
Remark 1 If the condition (C.1) replaces the condition (C.2) of Theorem 4.1 and the other conditions are not changed, then the conclusions of Theorem 4.1 cannot be guaranteed to be right as .
The example is as follows.
here is assumed to satisfy (A2.4) and be locally bounded, i.e., there is and with such that , , .
It is clear that r, f satisfy, respectively, (A2.1) and (A2.2), and F satisfies (A2.3), and (), i.e., satisfies (C.1) but it does not satisfy (C.2).
for every . Here is determined by , and .
, a.s. ().
There is finite value measurable function defined on such that , a.s. ().
for every . Here and are mutually independent.
Thus on by (4.9). Therefore is not an almost sure oscillation, and consequently, (4.6) is not almost surely oscillatory by Definition 2.3. □
This work was supported in part by the National Natural Science Foundation of China under Grants no. 11101054, Hunan Provincial Natural Science Foundation of China under Grant no. 12JJ4005 and the Scientific Research Funds of Hunan Provincial Science and Technology Department of China under Grants no. 2010FJ6036.
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