Subordinated Critical Branching Processes

• Subordinated critical branching processes are models combining BGW dynamics with random renewal time-changes and migration to modify reproduction timing and extinction rates.
• The framework reveals that emigration-dominated regimes lead to accelerated extinction, with survival probabilities decaying as distinct power laws.
• Subordination introduces memory effects and heavy-tailed phenomena, resulting in alternating regenerative cycles and unique conditional limit laws.

A subordinated critical branching process is a branching system, typically of Bienaymé–Galton–Watson (BGW) or continuous-state type, whose evolution is subjected to an additional stochastic time-change or “subordination”—usually achieved via renewal processes or subtractive subordinators. When such branching processes are further equipped with migration (immigration and/or emigration), subordination mechanisms induce nontrivial modifications in survival probabilities, limit distributions, and regeneration structure. Of special interest is the regime where emigration predominates. The following sections develop the theoretical framework, main asymptotic results, mathematical formulations, and implications for this class of models, with direct reference to the scenarios studied in (Yanev, 14 Sep 2025).

1. Subordination via Renewal Processes

In the canonical discrete-time BGW process with migration, subordination is implemented by replacing deterministic generation times with random epochs dictated by an independent renewal process. Formally, let $\{Z_n\}$ denote the BGW (+ migration) process and $\{N(t), t \geq 0\}$ the renewal process with interarrival times $J_i$ (i.i.d., nonnegative, nonarithmetic). The subordinated process is then

$Y(t) = Z_{N(t)}, \qquad t \geq 0,$

where $N(t) = \max\{ n : S_n \leq t \}$, $S_n = J_1 + J_2 + \cdots + J_n$, $S_0 = 0$. The process $Y$ is constant on each interval $[S_k, S_{k+1})$ and only jumps at renewal epochs. This time change introduces randomness in generation intervals, modeling environmental or operational uncertainty in reproductive timing.

2. Emigration-Dominated Regime: Asymptotic Behavior

Focus is on the regime where emigration dominates migration, formalized by the effective migration parameter

$$\theta = \frac{2 \mathbb{E}[M]}{\operatorname{Var}[X]} < 0,$$

with $M$ the net migration random variable per step, and $X$ the single-step BGW offspring variable.

Assume finite mean renewal time $\mu = \mathbb{E} J_1 < \infty$. Then, if the initial population is integrable, the non-extinction (survival) probability of the subordinated process $Y$ has the asymptotic form

$$\mathbb{P}\big( Y(t) > 0 \big) \sim L_\theta(t) \, (t/\mu)^{-(1 + |\theta|)}, \qquad t \rightarrow \infty,$$

where $L_\theta(t)$ is a slowly varying function. This scaling sharply contrasts the classical $O(1/t)$ Yaglom law of critical BGW without migration, and reflects accelerated extinction due to prevailing emigration.

If $Y(0)$ instead possesses a heavy tail, i.e.,

$$\mathbb{P}\left( Y(0) > x \right) \sim L_\gamma(x) \, x^{-\gamma}, \quad \gamma \in (0,1),$$

then long-term survival is governed by the initial data: $$\mathbb{P}\big( Y(t) > 0 \big) \sim L_{\theta,\gamma}(t) \, (t/\mu)^{-\gamma},$$ emphasizing a “memory effect” in the large-population initial condition.

If $\mu = \infty$ (the renewal law has infinite mean), the survival probability decays at a sublinear rate: $$\mathbb{P}\big( Y(t) > 0 \big) \sim \mathrm{const} \cdot t^{-\rho}, \quad \rho \in (0,1),$$ with possible further reduction to $t^{-\rho\gamma}$ in the heavy-tailed initial regime.

3. Conditional Limit Laws (Yaglom-Type Theorems)

With finite mean interarrivals ($\mu < \infty$) and when $Y(0)$ is integrable, the Yaglom-type conditional law for the normalized process is

$$\frac{Y(t)}{bt/\mu} \;\Big|\; \big\{ Y(t) > 0 \big\} \xrightarrow{\;d\;} \mathrm{Exp}(1)$$

as $t \to \infty$, where $2b = \operatorname{Var}[X]$. For infinite-mean or heavy-tailed initial populations, normalization shifts to sublinear rates (e.g., $t^\rho/L_\rho(t)$) and limit distributions may become non-exponential, with explicit Laplace transforms described in (Yanev, 14 Sep 2025).

4. Alternating Regenerative Processes

An “alternating regenerative branching” (ARB) process is constructed to allow the system to “restart” after absorption at zero. Down-times (sojourns at zero) and up-periods (regenerative cycles with positive population) are pieced together using two renewal processes, yielding a process $U(t)$ dynamic on $[0, \infty)$ that alternates between null and active phases.

The limiting distribution of the normalized $U(t)$, under appropriate conditions, emerges as a mixture:

• A continuous part, supported on $(0, \infty)$, identified via products of $V$ (limit law for the “up” process) and $W$ (Beta-distributed fractions of cycle times),
• An atom at zero, proportional to the frequency of down-periods.

In heavy-tailed initial populations or infinite mean cases, explicit formulas for the limiting mixture are provided, involving normalization by $t^\rho/L_\rho(t)$ and randomization from underlying renewal epochs.

5. Comparison to Immigration-Dominated and Classical Settings

Prior works focus on migration-dominated by immigration ($0 < \theta < 1$), where survival decay is slower and norming often differs (e.g., gamma-type limits for positive recurrent processes). In the emigration-dominated ($\theta < 0$) and subordinated context studied here, time-changes via the renewal process induce faster extinction, reduce survival exponents, and modify the normalization in Yaglom laws to sublinear or heavy-tailed, especially when both the initial distribution and the renewal law are heavy-tailed.

Classical Yaglom theory is recovered only in the absence of migration, finite mean interarrivals, and integrable initial data. The renewal-induced subordination thus acts as a mechanism interpolating between memoryless (Markovian) behavior and memory-rich, heavy-tailed phenomena, affecting extinction rates, population resurgence frequencies, and the long-run distribution profiles.

Summary: Key Theoretical Formulations

Scenario	Survival Probability Asymptotics	Conditional Limit Law
Finite mean, integrable initial	$L_\theta(t) (t/\mu)^{-(1 + |\theta|)}$	$\frac{Y(t)}{bt/\mu} \to \text{Exp}(1)$
Finite mean, heavy-tailed initial $\gamma$	$L_{\theta,\gamma}(t) (t/\mu)^{-\gamma}$	Limit law with explicit Laplace transform
Infinite mean interarrivals ($\mu = \infty$)	$\text{const} \cdot t^{-\rho}$ (or $t^{-\rho\gamma}$)	Normalization by $t^\rho/L_\rho(t)$
Alternating regenerative (ARB) process	Mixture (atom at 0 + continuous law on $(0,\infty)$)	Involves Beta randomization, product distributions

Here, $L_\theta(\cdot)$ and $L_\gamma(\cdot)$ are slowly varying, $b$ is half the offspring variance, and $\rho \in (0,1)$ is determined by the renewal law.

The mathematical constructs, asymptotic estimates, and probabilistic limit theorems summarized above are all stated precisely in (Yanev, 14 Sep 2025), which extends the classical theory of critical BGW processes and migration to subordinated—renewal time-changed—settings, with special attention to prevailing emigration. For the role of $\theta$ in migration processes, see also (Yanev, 2014). For classical Yaglom theory and time-changed branching, see (Kersting, 2021), and for analytic properties of related coalescence and extreme-value phenomena under subordination, see (Foucart et al., 2016, Foucart et al., 2018), and (Felipe et al., 2016).

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This comprehensive account describes subordinated critical branching processes with prevailing emigration, their asymptotic regimes, normalizations, limit theorems, and the regenerative structures introduced by combined migration and subordination effects. All claims, formulas, and scenarios correspond precisely to those established in (Yanev, 14 Sep 2025).