Large Deviations for the Supremum of Partial Sums of Non-Independent and Non-Identically Distributed Random Variables

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.