Lemma 2.1 [6]
Let be a sequence of arbitrary random variables. Let and , . Let be a sequence of non-negative even functions of x,
(1)
and let be an increasing sequence of positive numbers. If
(2)
and , then ,
(3)
Lemma 2.2 Let be the m-dimension stock relative price sequence as given above and be the m-dimension investment strategy sequence as given above. Suppose that is measurable with respect to and is defined as in Lemma 2.1. Let , if
(4)
then ,
(5)
where , .
Proof Because is a stochastic adapted sequence, from Lemma 2.1, this lemma holds. □
Definition 2.3 Suppose that is m-dimension vector sequence and . If there exists a sequence of numbers that satisfies , such that when , , , , we have
(6)
then is called a ∗-mixing sequence.
We easily see that (3) is equivalent to the following equation: ,
(7)
Lemma 2.4 Let be a ∗-mixing sequence defined by Definition 2.3. Suppose 1-dimension random variable and , then ,
Proof The proof procedure is almost similar to [7], p.139. □
Theorem 2.5 Let be an m stock relative price sequence and a ∗-mixing sequence. is the optimized investment strategy sequence. Suppose that , and , . If (4) holds, then
(8)
Proof From Lemma 2.2, we know that ,
(9)
then from Lemma 2.4, we have
(10)
Since when (), we can choose m such that is sufficiently small, from (9) and (10), we know that (8) holds. □
Remark 2.6 In fact, a ∗-mixing sequence means that the sequence is gradually independent. In this paper, we suppose the relative price sequence satisfying the ∗-mixing condition, which means that the relative stock prices in two periods gradually become independent when the interval between two periods is longer and longer. In the last section, we will give two numerical examples, which show that a market satisfying the ∗-mixing condition exists. Therefore, Theorem 2.5 is meaningful.
Remark 2.7 In Theorem 2.5, we suppose that is measurable with respect to . It means that the investment strategy in the n th period is totally decided by the information of the stock prices of the N periods before it. Because we consider the long term behavior of a sequence investment, this assumption is meaningful for the financial market.
Remark 2.8 The equality (8) shows that the average return of the long term behavior of a sequence investment converges to the average of the expectation return in every period in probability 1 under the conditions in Theorem 2.5.
Corollary 2.9 Let be the m stock relative price sequence and be the ∗-mixing sequence. Let be the optimized investment strategy sequence. Suppose that , and , , where is a constant. If
(11)
then
(12)
Proof Letting in Theorem 2.5, this corollary follows. □
Theorem 2.10 Let and be defined as in Theorem 2.5 and let condition (1) in Lemma 2.1 be replaced by the following condition: as increases,
(13)
If (4) holds, then
(14)
Proof Let and . By (13), we have
So
(15)
In addition, from (13),
thus, according to the Borel-Cantelli lemma, we have
(16)
By (15) and (16), we find that the conclusion holds. □