Tauberian theorems of exponential type and its applications to probability theory

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・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.