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Tauberian theorems of exponential type and its applications to probability theory

指数型陶伯定理及其在概率论中的应用

基本信息

批准号：
13640104
负责人：
KASAHARA Yuji
金额：
$ 2.11万
依托单位：
Ochanomizu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

・A theorem that treats such relationship is called a Tauberian theorem. In our research we found that a Tauberian theorem of exponential type is essentially equivalent to the inverse problem for the Fenchel-Legendre transformation. We also obtained a condition for the latter problem. The result is useful when we treat functions without assuming smoothness.・We also found that the same idea is applicable to the following problem : Limit theorems for sums of independent, identically distributed random variables are classical and it known that we need non-linear normalization if the tail probability is very heavy. Such cases appears, for example, in excursion intervals of two-dimensional random walk. We proved that in such cases the sum has an asymptotic expansion using order statistics.・It is well known that the time a Brownian motion spends on the positive side obeys the arc-sine law. We studied similar results for more general diffusions. Although we cannot write down the explicit law of the time spent on the positive side, we obtained the relationship between the asymptotic behavior around 0 of the distribution function and that of the speed measure. We next obtained results on the asymptotic behavior of the distribution function of the time spent on the positive side in the case where the speed measure increases in exponential order. Our proof is based on the idea we used in the theory of Tauberian theorems of exponential type.

·处理这种关系的定理称为陶伯定理。在我们的研究中，我们发现指数型的Tauberian定理本质上等价于Fenchel-Legendre变换的反问题。我们还得到了后一个问题的一个条件。当我们处理函数而不假设光滑性时，该结果很有用。·我们还发现，同样的想法是适用于以下问题：极限定理的总和独立，同分布的随机变量是经典的，它知道，我们需要非线性归一化，如果尾部概率是非常沉重的。例如，这种情况出现在二维随机游走的偏移区间中。我们使用顺序统计量证明了在这种情况下，和具有渐近展开式。·众所周知，布朗运动在正侧所花费的时间服从反正弦定律。我们研究了更一般的扩散类似的结果。虽然我们不能写出在正侧所花费的时间的明确规律，但我们得到了分布函数在0附近的渐近行为与速度测度在0附近的渐近行为之间的关系。接下来，我们得到了在速度测度以指数顺序增加的情况下，在正侧花费的时间的分布函数的渐近行为的结果。我们的证明是基于我们在指数型的Tauberian定理理论中使用的思想。