In this paper, we establish the large deviation principle for the supremum of partial sums of a sequence of non-independent and non-identically distributed random variables. Let Xn: n≥1 be a sequence of random variables with the same negative mean, and denote Sn=∑i=1nXi, S0=0, and assume that the limit Λ (θ) =limn→∞1nΛn (θ) of the logarithmic moment generating functions Λn (θ) =logEeθSn exists and is essentially smooth and lower semi-continuous. We prove that the sequence 1lsupn≥0Sn: l≥1 satisfies the large deviation principle on R and provide the exact form of its rate function. As a consequence, we obtain the large deviation principle for the supremum of sums formed by combining independent and identically distributed components with correlated components. As applications, we analyze the first-order autoregressive process and the Poisson–Gaussian mixture case and derive exact expressions for the corresponding rate functions and asymptotic estimates for the decay of tail probabilities.
Wang et al. (Wed,) studied this question.