If X n are i.i.d. random variables, let S n = X 1 + ⋯ ⋯ + X n be the associated random walk and M ( ξ ξ ) the moment generating function of X 1 . For ξ ξ ∈ ∈ R , E [ e ξ ξ S n ] = M ( ξ ξ ) n . Hence, by Markov's inequality, one has for ξ ξ ≥ ≥ 0 and a ∈ ∈ R P ( S n / n > a ) ≤ ≤ e − − n ξ ξ a M ( ξ ξ ) n = exp ⁡ ⁡ [ − − n ( ξ ξ a − − Λ Λ ( ξ ξ ) ) ] where Λ Λ ( ξ ξ ) = log ⁡ ⁡ M ( ξ ξ ) . Since the left-hand side is independent of ξ ξ , we may take the infimum of the right-hand side, which leads one to consider the supremum of ξ ξ a − − Λ Λ ( ξ ξ ) , i.e., the Legendre transform of Λ Λ , evaluated at x = a .