“…As discussed earlier, choosing W n ≡ δ 0 gives P n,p,Wn = C n,p , and W n ≡ Exp(1) yields P n,p,Wn = U n,p ; in both cases for W n ∼ W n we have W n /n → 0 in probability and we recover [10,Theorem C]. Hence, we can see that C n,p and U n,p share the same LDP behaviour in high dimensions, which is in line with similar observations made for other functionals (see, e.g., [1,12,14]). Moreover, the result even implies a certain universality of the rate function, since despite the expected sensitivity of LDPs to the underlying distributions, the rate function is the same for all sequences (W n ) n∈N that share the same limiting behaviour.…”