Principle of the largest exponential term
= Principle of the largest exponential term
{title2=$\limsup_La_L^{-1}\log\sum_{j=1}^rp_{j,L}=\max_j\limsup_La_L^{-1}\log p_{j,L}$}
For a fixed finite number of nonnegative terms and $a_L\to\infty$, use $\max_jp_{j,L}\le\sum_jp_{j,L}\le r\max_jp_{j,L}$. The <logarithm> of $r$, divided by the <large-deviation speed>, tends to zero, and a finite maximum commutes with the <limit superior>. If all individual logarithmic limits exist, the limit of the sum is their maximum.