Functional limit theorems for sums of independent random variables with random coefficients
Let \(\eta_n\), \(n\in N\), be i.i.d. random variables belonging to the domain of normal attraction of symmetric \(p\)-stable random variables \(\gamma_p\), \(0<p\leq 2\). Consider the sums \(S_n= \sum^{k(n)}_{i=1} \xi_{ni} \eta_i\), where the random coefficients \(\xi_{ni}\) are independent of \(\{\eta_n\}\). Conditions under which the convergence in distribution of \(S_n\) is equivalent to the convergence in distribution of \(T_n= \sum^{k(n)}_{i=1} |\xi_{ni} |^p\) are found. One of the results states that if \(\xi_{ni}\) are infinitesimal and \(\eta_n\) are symmetric for \(p\leq 1\) or centered for \(p>1\), then \(S_n\) converges in distribution to \(s\gamma_p\) for some positive \(s\) iff \(T_n\) converges in distributon to \(s^p\). The functional limit theorem for corresponding step functions is also obtained.
scientific article; zbMATH DE number 1465958
scientific article