受控分支过程的大偏差：随机斜率、调和矩与转移原理

中文摘要 AI 辅助

设 \\(\{X_n:n\ge0\}\\) 为具有控制 \\(\phi_n\\) 的受控分支过程，并设 \\(N_n=\phi_n(X_n)\\) 为其祖代计数。我们在超临界和临界状态下，研究 \\(X_{n+1}/N_n\\)、\\(N_n/X_n\\) 和 \\(X_{n+1}/X_n\\) 在其自然正分母事件上的大偏差。我们处理世代范围的随机斜率和渐近确定的一阶斜率。对于渐近仿射的随机斜率控制，仿射递归和随机环境乘积产生了精确的 Bahadur--Rao--Petrov 上尾渐近。正的 Cramér 倾斜通过非平凡归一化极限识别出前因子，并且在上偏差事件条件下，三个比率集中在其环境中心附近。对于个体和控制，该过程是带移民的随机环境中的精确分支过程（BPREI）。受此联系启发，我们建立了 BPREI 的精确上大偏差渐近，并将其正变换和调和矩特征转移到受控过程。在非降超临界情形下，调和矩表现出精确的环境-边界-持久性三分法。在中心化有限方差临界类中，有限时域最小值方法对所有正调和阶和三个比率偏差概率产生了精确的 \\(n^{-1/2}\\) 尺度极限。对于确定性一阶斜率，环境乘积机制消失。匹配的下、上概率生成函数包络将辅助精确递归特征转移到没有精确分支递归的控制，产生不同的超临界和临界调和矩三分法。因此，一阶随机性的位置决定了下尾机制，而移民可能改变精确常数、相边界或衰减指数。

英文摘要

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.

发表机构

• University of Extremadura（埃斯特雷马杜拉大学）
• George Mason University（乔治梅森大学）

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