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

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640104/
• 关键词: Tauberian theorem; Laplace transform; Legendre transform; ブラウン運動; 逆正弦法則; Bessel過程; 拡散過程; Fenchel-Legendre変換

・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，我们发现指数型陶伯定理本质上等同于芬切尔-勒让德变换的反问题，我们还得到了后一个问题的条件。当我们在不假设平滑性的情况下处理函数时很有用。・我们还发现相同的想法适用于以下问题：独立同分布随机变量之和的极限定理是经典的，并且知道我们需要如果尾部概率非常大，则进行非线性归一化。例如，在二维随机游走的偏移区间中，我们证明了在这种情况下，总和具有阶统计量的渐近展开。·这是众所周知的。布朗运动在正侧花费的时间遵循反正弦定律我们研究了更一般的扩散的类似结果，尽管我们无法写出在正侧花费的时间的明确定律，但我们获得了之间的关系。周围的渐近行为接下来，我们获得了在速度测量按指数顺序增加的情况下，分布函数在正侧的渐近行为的结果。我们在指数型陶伯定理理论中使用的想法。