Estimates for the distribution of sums and maxima of sums of random variables without the Cramér condition
Let \(X_1,X_2,\dots\) be identically distributed independent random variables with distribution function \(F(t)\), and let \[ S_n=\sum_{j=1}^n X_j,\quad \overline S_n(a)=\max_{k\leq n}(S_k-ka). \] It is assumed that the tails \(F(-t)=\mathbf{P}\{X_j<-t\}\) and \(1-F(t)=\mathbf{P}\{X_j\geq t\}\) are majorized or minorized by a regularly varying or semiexponential function. The main results concern the upper and lower bounds for the probabilities \(\mathbf{P}\{S_n>x\}\) and \(\mathbf{P}\{\overline S_n(a)>x\}\). Various cases are considered. Among them are the cases in which the mean value of \(X_1\) does not exist, the variance does not exist, etc. The main technical tools are a properly chosen truncating procedure and exponential bounds of Chebyshev type. Necessary and sufficient conditions are also presented under which the uniform relative convergence to a stable law holds. The article is concluded with the laws of iterated logarithm in the case when the second moment is infinite.
scientific article; zbMATH DE number 1603201
scientific article