Large Deviations for Controlled Branching Processes: Random Slopes, Harmonic Moments, and Transfer Principles
Let \(\{X_n:n\ge0\}\) be a controlled branching process with controls \(ϕ_n\), and let \(N_n=ϕ_n(X_n)\) be its progenitor count. We study large deviations of \(X_{n+1}/N_n\), \(N_n/X_n\), and \(X_{n+1}/X_n\), on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-wide random slopes and asymptotically deterministic first-order slopes. For asymptotically affine random-slope controls, an affine recursion and a random environmental product yield a sharp Bahadur--Rao--Petrov upper-tail asymptotic. A positive Cramér tilt identifies the prefactor via a nontrivial normalized limit, and conditional on the upper-deviation event the three ratios concentrate around their environmental centers. With individual-sum controls, the process is an exact branching process in a random environment with immigration (BPREI). Motivated by this connection, we establish sharp upper large-deviation asymptotics for BPREI and transfer its positive-transform and harmonic-moment profiles to the controlled process. In the nondecreasing supercritical case, harmonic moments exhibit an exact environmental--boundary--persistence trichotomy. In a centered finite-variance critical class, finite-horizon-minimum methods yield exact \(n^{-1/2}\)-scale limits for all positive harmonic orders and for the three ratio-deviation probabilities. For deterministic first-order slopes, the environmental-product mechanism disappears. Matched lower and upper probability-generating-function envelopes transfer auxiliary exact-recursion profiles to controls without an exact branching recursion, producing distinct supercritical and critical harmonic-moment trichotomies. Thus the location of first-order randomness determines the lower-tail mechanism, while immigration may change a sharp constant, a phase boundary, or the decay exponent.