Before revision:   We have different types of results according to the magnitude relationship of thickness of tails of L and S. Especially if the tail of S is enough thinner than the one of L, then  VaR(L+S) - VaR(L) is asymptotically equivalent to an expected loss of S when L and S are independent .
After revision:  We have different types of results according to the magnitude relationship of thickness of tails of L and S. Especially if the tail of S is sufficiently thinner than that of L, then VaR(L+S) - VaR(L) is asymptotically equivalent to an expected loss of S when L and S are independent .