In their paper The Maximum of independent Geometric Random Variables as the Time for Genomic Evolutionthe authors noted that if to consider the genomic word of L letters, than the measure of the time it takes to obtain a successful genomic word $R_L=max{X_1, \ldots, X_L}$. Where $X_i, i=1, \ldots, L$ are independent geometric random variables.
My questions is, what, in terms of genomic would mean an estimate of $E(\sum_{i=1}^La_iX_i)^{2n}$, where $a_i\in R$ are weights?