Cramér Moderate Deviations for a Supercritical Galton–Watson Process with Immigration

Abstract

Consider a supercritical Galton–Watson process with immigration (Xn;n≥0). The Lotka–Nagaev estimator Xn+1Xn is a common estimator for the offspring mean. In this work, we used the Martingale method to establish several types of Cramér moderate deviation results for the Lotka–Nagaev estimator. To satisfy our needs, we employed the well-known Cramér approach for our proofs, which establishes the moderate deviation of the sum of the independent variables. Simultaneously, we provided a concrete example of its applicability in constructing confidence intervals.