Principle of the largest exponential term (${\mathrm{limsup}}_L{a}_L^{-1}\log {\sum }_{j=1}^r{p}_{j,L}={\max }_j{\mathrm{limsup}}_L{a}_L^{-1}\log p_{j,L}$)

For a fixed finite number of nonnegative terms and $a_L\to \infty$, use ${\max }_j{p}_{j,L}\le {\sum }_j{p}_{j,L}\le r{\max }_j{p}_{j,L}$. 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.