ASYMPTOTIC NORMALITY OF NUMBER OF CROSSINGS OF LEVEL ZERO BY A GAUSSIAN PROCESS
ASYMPTOTIC NORMALITY OF NUMBER OF CROSSINGS OF LEVEL ZERO BY A GAUSSIAN PROCESS
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DOI：
10.1137/1114035
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发表时间：
1969-01-01
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影响因子：
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通讯作者：
MALEVICH, TL
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文献类型：
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作者：
MALEVICH, TL
由于变量G（a，B）是[a，B]上的一个可加函数，所以可以用推导随机变量和的极限定理的方法来研究它的渐近分布（如B a）.特别地，当Dqx（0，T）fT，fl> 0（7]，8]）时，不难证明qx（0，T）在强混合条件下作为T渐近正态。然而，已经假设，因为过程x（t）是高斯的，所以中心极限定理对qx（0，T）（在Dr/（0，T）的适当增长下）成立，而不必假设强混合条件。以下情况证实了这一说法。
Since the variable G (a, b) is an additive function on [a, b], it is possible to study its asymptotic distribution (as b a) by the methods used to deduce limit theorems for sums of randomvariables. In particular, it is not hard to show that qx (0, T) is asymptotically normal as T under strong mixing conditions when Dqx (0, T) fiT, fl> 0 (7], 8]). However, it has been hypothesized that because the process x (t) is Gaussian, the central limit theorem holds for qx (0, T)(under suitable growth ofDr/(0, T)) without having to assume the strong mixing condition. This assertion is confirmed by the following.