Upper and lower bounds in exponential Tauberian theorems (Q991572)

The idea of this paper ``Tauberian theorems can be used to obtain simple large deviation results.'' That's why for r.v. \(X\geq 0\) on a probability space \((\Omega,\mathcal{A},P),\) an event \(A\in \mathcal{A}\) with \(P(A)>0\) and \(\alpha\in (0,1),\;\beta>0\) with \(\frac 1{\alpha}=\frac 1{\beta}+1\) the author at first derives from De Bruijn's Tauberian theorem Theorem 1.1. The limit \(r\) exists if and only if the limit \(s\) exists, \[ r=\lim_{\lambda\to \infty}\lambda^{-\alpha}\log E(e^{-\lambda X}\cdot 1_A)\leq 0,\quad s=\lim_{\varepsilon\to 0}\varepsilon^{\beta}\log E(X\leq \varepsilon, A)\leq 0, \] and in this case we have \(|\alpha r|^{1/\alpha}=|\beta s|^{1/\beta}.\) Then in section 2, it is shown that Theorem 1.1 can be used to derive a large deviation principle for Brownian motion. But in general the limit \(s\) does not necessarily exists. And for large deviation results one usually considers upper and lower limits. That's why in s. 3 in the same conditions an analogue of Theorem 1.1 is derived (its proof uses only elementary methods): Theorem 3.1. a) The upper limits \[ \bar{r}=\limsup_{\lambda\to \infty}\lambda^{-\alpha}\log E(e^{-\lambda X}\cdot 1_A),\;\;\text{and}\;\;\bar{s}=\limsup_{\varepsilon\to 0}\varepsilon^{\beta}\log E(X\leq \varepsilon, A) \] satisfy \(|\alpha \bar{r}|^{1/\alpha}=|\beta \bar{s}|^{1/\beta}.\) b) The lower limits \[ \underline{r}=\liminf_{\lambda\to \infty}\frac 1{\lambda^{\alpha}}\log E(e^{-\lambda X}\cdot 1_A),\;\;\text{and}\;\;\underline{s}=\liminf_{\varepsilon\to 0}\varepsilon^{\beta}\log E(X\leq \varepsilon, A) \] satisfy \(|\alpha \underline{r}|^{1/\alpha} \leq |\beta \underline{s}|^{1/\beta} \leq |e^{H(\alpha)}\alpha \underline{r}|^{1/\alpha},\) \(H(\alpha)=-\alpha \log \alpha-(1-\alpha)\log(1-\alpha),\) and both bounds are sharp.