A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables

The author obtains that the asymptotic relationsbegin{align*}mathbb{P}Big(sum_{i=1}^n theta_iX_i >xBig)simmathbb{P}Big(max_{1leq mleq n}sum_{i=1}^m theta_iX_i>xBig)sim mathbb{P}Big(max_{1leq ileqn}theta_iX_i>xBig)sim sum_{i=1}^n {mathbb{P}( theta_iX_i>x)}end{align*}hold as $xtoinfty$, where the random weights$theta_1,cdots,theta_n$ are bounded away both from $0$ and from$infty$ with no dependency assumptions, independent of the primaryrandom variables $X_1,cdots,X_n$ which have a certain kind ofdependence structure and follow non-identically subexponentialdistributions. In particular, the asymptotic relations remain truewhen $X_1,cdots, X_n$ jointly follow a pairwise Sarmanovdistribution.