Functional Erdős-Rényi laws for semiexponential random variables
Let \(Y, Y_1, Y_2,\dots{}\) be i.i.d. with \({\mathbb E}Y=0\) and \({\mathbb E} \exp\{\lambda Y\}<\infty\) for all \(\lambda\leq 0\). Assume that \[ a_1(t) \exp\{-b(t)t^r\}\leq {\mathbb P}(Y\geq t)\leq a_2(t)\exp\{-b(t)t^r\} \] for \(t\) large enough, where \(0<r\leq 1\), \(a_1,a_2,b\) are slowly varying functions and \(b(t)/t^{r-1}\) is nonincreasing and \(b(t)\to 0\) if \(r=1\). Let \(E=\{x=x(t)\in L^1[0,1]: x(0)=0\}\) and let be the metric on \(E\) given by \(d(x,y)=\int_0^1|x(s)-y(s)|ds\). Equip \(E\) with its Borel \(\sigma\)-field \({\mathcal B}\). Let \[ Z_n(t)= \frac{1}{n}\sum_{j=1}^{[nt]}Y_j-(t- [nt]/n)Y_{[nt]+1}, \quad 0\leq t \leq 1. \] The following theorems are the main results of the paper. Theorem 1. Assume \(0<r<1\). The distributions of \(Z_n\) satisfy a large deviation principle on \((E,d)\) with normalization \(b(n)n^r\) and with the good rate function \(I(x)=\sum_{t: x(t^+)\not = x(t^-)}(x(t^+)-x(t^-))^r, x\) nondecreasing pure jump function, \(I(x)=\infty\) otherwise. Theorem 2. Assume \(r=1\). The distributions of \(Z_n\) satisfy a large deviation principle on \((E,d)\) with normalization \(b(n)n\) and with the good rate function \(\widetilde{I}(x)=x(1), x\) nondecreasing function, \(\widetilde I(x)=\infty\) otherwise. The author derives the ``Strassen-type'' theorems from Theorems 1 and 2.
scientific article; zbMATH DE number 1355239
scientific article