Tauberian theorems of exponential type
Tauberian theorems of exponential type
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指数型陶伯定理
DOI：
10.1215/kjm/1250522571
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发表时间：
1978
影响因子：
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通讯作者：
Y. Kasahara
中科院分区：
文献类型：
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作者：
Y. Kasahara
由Karamata引入的正则变函数的概念大大扩展了Hardy-Littlewood Tauberian定理并简化了其证明。根据Karamata的Tauberian定理，非减函数a（t）在0处规则变化，当且仅当它的拉普拉斯变换F（t）在ω处规则变化（见[2]或[ 9 ]）。然而，他的方法在a（t）或F（A）以指数顺序变化的情况下为我们提供的信息很少（参见图1）。[3]）。这种情况在概率论中的一些问题中是很有意义的，Varadhan [10]和福岛[3]等曾研究过这种情况。许多作者也研究过类似的问题。L. Davies [1 ]和永井[7 ]（或[4 ]）研究了a（t）作为co与F（t）作为s -> -co的渐近行为之间的关系，Davies [1 ]和Kôno [5]处理了用矩代替拉普拉斯变换的情况。本文的目的是给出一个最一般形式的Tauberian定理。第一节给出了主要定理及其证明。在第二节中，我们将它应用于各种情况，并看到上面提到的Tauberian定理作为我们定理的特殊情况而得到。
The notion of regularly varying functions, which was introduced by Karamata, extended greatly the Hardy-Littlewood Tauberian theorem and simplified its proof. According to Karamata's Tauberian theorem, a nondecreasing function a(t), varies regularly at 0, if and only if its Laplace transform F()) varies regularly at co (see [2] or [ 9 ] ) . However, his method provides us with little information in a case where a(t) or F (A) varies in an exponential order (cf. [3]). Such a case is of interest in some problems in probability theory and studied by Varadhan [10] and by Fukushima [3] etc. Similar problems have been studied by many authors. L. Davies [1 ] and Nagai [7 ] (or [4 ]) studied the relation between the asymptotic behaviour of a(t) as co and that of F()) a s —> — co. D avies [1 ] and Kôno [5] treated the case where the Laplace transform is replaced by the moments. The aim of this paper is to give a Tauberian theorem in a most general form. In section 1 the main theorem is stated with its proof. In section 2, we apply it to various cases and see that the Tauberian theorems mentioned above are obtained as special cases of our theorem.