On Chung’s Law for Nonidentically Distributed Independent Summands
On Chung’s Law for Nonidentically Distributed Independent Summands
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DOI：
10.1137/1131065
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发表时间：
1987-09
影响因子：
0.6
通讯作者：
A. Martikainen
中科院分区：
文献类型：
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作者：
A. Martikainen
如果我们将(7)中的指数1，2替换为2，1，我们得到E{e2(t2-t1)}的类似表达式。现在，定理的断言立即从(6)开始。举例说明。在方程(1)-(2)中，设n=d1，yi(T)=a+wt，q=(0，1)。然后，很明显，v，(Y)y(1y)，dv/dy 2y和elt1t21-<；elaA2L。在书[2]中，详细研究了漂移系数和扩散系数变化时过程的首次退出时间的行为。对于非齐次过程和T，T2有界时间，在[3]中导出了类似于(5)的估计，并用它证明了在有界区域中受控扩散过程的“随机极大值原理”。对于任意形式的强马氏过程，文[5]中给出了(5)型的估计(然而，文中所作的假设对于非扩散过程是限制性的，不容易检验)。
If we replaced the indices 1, 2 in (7) by 2, 1, we get an analogous expression for E {e2 (T2-T1)}. Now the assertion of the theorem follows at once from (6). Example. In equations (1)-(2) let n= d 1, yi (t)= a+ wt, Q=(0, 1). Then, obviously, v,(y) y (1 y), dv/dy 2y and ElT1 T21-< Ela a2l. The behavior of the first exit times of the process when the drift and diffusion coefficients vary has been studied in detailin the book [2]. However, it solves problems of a different sort than the one considered above.For nonhomogeneous processes and T, T2 bounded timesan estimate similar to (5) was derived in [3] and used to prove the" stochastic maximum principle" established in [4] for a controlled diffusion process ina bounded region. For strong Markov processes of arbitrary form, an estimate of type (5) was derived in [5](however, the assumptions-made there are restrictive and not easy to check for nondiffusional processes).