A large deviation principle for weighted sums of independent identically distributed random variables
Let \(X_0,X_1, \dots\) be i.i.d. random variables such that \(EX_0 =0\), \(EX^2_0=1\) and \(M(t)= E(\exp\{X_0\}) <\infty\), \(|t|< \infty\). Let \(a_j(\lambda)\), \(j=0,1,2, \dots\), \(\lambda >0\), where \(\lambda\) is a continuous parameter or \(\lambda\in N\), be real numbers such that for \(\nu=1\) and hence for each \(\nu=1,2, \dots\), \(\sum^\infty_{j=0} |a_j(\lambda) |^\nu <\infty\) for all \(\lambda\). The authors considers the following weighted means of the sequence \(\{X_n,n\geq 0\}\): \(V(\lambda)= \sum^\infty_{j=0} a_j(\lambda)X_j\). Starting from Cramer's theorem on large deviations of the Cesàro means \((X_0+ X_1+ \cdots+ X_{n-1})/n\), the authors show that other weighted sums \(V(n)\) exhibit a similar large deviation behaviour if the weights satisfy certain regularity conditions.
independent identically distributed random variables
scientific article; zbMATH DE number 1556952