Critical Finite Variance Branching Processes

• Critical finite variance branching processes are stochastic models where each individual produces offspring with mean one and finite variance, ensuring eventual extinction without conditioning.
• Recent studies reveal that rescaled genealogical trees converge to the Brownian Continuum Random Tree and display universal Yaglom-type exponential laws under survival conditioning.
• Techniques such as spine decompositions, martingale convergence, and generating function methods yield practical insights across models like Galton–Watson, Bellman–Harris, and branching in random environments.

A critical finite variance branching process is a stochastic population model where each individual produces a random number of offspring according to a critical (mean one) offspring law with finite variance. This class, which includes the classical Galton–Watson process, Bellman–Harris processes, their continuous-state analogues, and models with random or varying environments, displays universal large-time behavior when conditioned on survival or rare population-size events. Recent research has established detailed invariance principles, genealogical structure results, and limit theorems for these processes in great generality, notably showing convergence to the @@@@1@@@@ (CRT), universal Yaglom-type exponential laws, and explicit genealogical coalescent structures under various conditioning regimes.

1. Model Classes and Finite Variance Criticality

Critical finite variance branching processes are characterized by the property that the mean number of offspring per individual equals one (criticality), and the variance of the offspring distribution is strictly positive and finite. This defining feature appears across multiple frameworks:

• Galton–Watson processes: Discrete-time, single-type branching with i.i.d. offspring numbers; criticality requires $\mathbf{E}[\xi] = 1$, and finite variance $\mathbf{E}[(\xi-1)^2] = \sigma^2 \in (0, \infty)$ (Kersting, 2017).
• Bellman–Harris processes: Continuous-time with random i.i.d. lifetimes; criticality and variance as above, with added structure from the lifetime law (Hong et al., 2018).
• Crump–Mode–Jagers processes: Age-dependent, possibly overlapping generations, generalized settings in which individuals give birth at possibly non-constant times throughout their lives (&&&2&&&).
• Continuous-state branching processes (CSBP): Population size evolves as a positive-valued process with branching mechanism $\psi(\lambda) = \lambda^2 L(1/\lambda)$ for some slowly varying $L(\cdot)$, finite variance if $L(x) \to \sigma^2/2$ as $x \to \infty$ (Ren et al., 2013).
• Branching in random or varying environment: Offspring law may vary in time or depend on a random environment process, with corresponding finite-variance criteria (Grama et al., 2024, Vatutin et al., 22 Jun 2025, Kersting, 2017).

Criticality guarantees (almost sure) extinction of the process, while finite variance conditions ensure Brownian universality in fluctuations and scaling limits.

2. Universal Scaling Limits and the Brownian Continuum Random Tree

Across a variety of critical finite variance branching processes, a central result is that the rescaled genealogical tree, conditioned on survival to large time or size, converges in the Gromov–Hausdorff–(weak) topology to the Brownian Continuum Random Tree (CRT):

• General invariance principle: For branching Markov processes in general Lusin spaces with finite variance, genealogical metric measure spaces under suitable conditioning and $n$-scaling converge to the CRT coded by a Brownian excursion of variance $\sigma^2$ (Horton et al., 9 Jan 2026). The convergence is in the Gromov–Hausdorff–weak topology, and the limit is universal, depending only on variance-like parameters derived from the process.
• Branching diffusions in bounded domains: Critical branching diffusions in bounded subdomains of $\mathbb{R}^d$ exhibit the same scaling limit: genealogies rescaled by $1/t$ at large time $t$ converge to the CRT (Powell, 2015).
• Genealogy coding: Trees are coded via height processes (contour/DFS exploration), with the exploration process converging to a Brownian excursion. The CRT’s metric structure is characterized by $$d_e(s,t) = e(s) + e(t) - 2 \min_{u \in [s \wedge t, s \vee t]} e(u)$$, where $e$ is a Brownian excursion (Horton et al., 9 Jan 2026, Powell, 2015).

This scaling limit reveals the universality class of critical finite variance branching structures, showing that genealogical measures lose memory of branching details, preserving only mass and variance information.

3. Classic Yaglom Law and Conditional Population Size Limits

A central feature of critical finite variance branching is the universal behavior of the population size conditioned on survival up to large time $n$ or $t$:

• Yaglom-type results: In the Galton–Watson, Bellman–Harris, CSBP, and varying/random environment settings, classic results show

$$\lim_{n \to \infty} \mathbb{P}\left(\frac{Z_n}{n \sigma^2/2} \leq y\ \big|\ Z_n>0\right) = 1 - e^{-y},\qquad y \ge 0$$

and analogously for continuous time with $t$ replacing $n$ (Kersting, 2017, Liu et al., 2018, Ren et al., 2013, Hong et al., 2018).

• Survival probability decay: The probability of survival decays as $\sim 2/(\sigma^2 n)$ in discrete time and as $2/(\sigma^2 t)$ in continuous time (Kersting, 2017, Ren et al., 2013, Hong et al., 2018).

This law dictates that, upon survival for many generations, the process is “rare” and its size becomes asymptotically exponential under $t$- or $n$-scaling, with a parameter set by the offspring variance.

4. Genealogical Structures and Reduced Processes

The genealogical structure “pruned” by conditioning on small populations or survival exhibits explicit, coalescent-like or Markovian behavior:

• Reduced critical processes: The number of ancestral lineages at an earlier time $m$ that survive to time $n$ in the Galton–Watson or Bellman–Harris process, $Z(m, n)$, forms a reduced process with explicit limiting distributions (Liu et al., 2018, Hong et al., 2018).
• Small population windows: Conditioning on $0 < Z(n) \leq \varphi(n)$, with $\varphi(n)$ sublinear or linear in $n$, yields, for the reduced process, limit laws such as exponential-Erlang mixtures (for sublinear windows) or beta-exponential mixtures (for linear windows), and explicit distributional limits for the most recent common ancestor distance.
• Pure death process “coming down from infinity”: In age-structured (Crump–Mode–Jagers) models, Sagitov identifies the post-survival genealogy as a pure death process started from infinity, with death rates depending on offspring variance, mean generation length, and lifetime tails (Sagitov, 2021).
• Genealogical metrics: For measure-valued or nonlocal branching, the “exploration measure” and scaled tree metrics are used; for spatial branching, the CRT emerges when genealogical distances and masses are properly normalized (Horton et al., 9 Jan 2026, Powell, 2015).

A key implication is that rare (conditioned) critical populations are highly coalescent, with a small number of ancestral lineages at any sublinear (or “recent” linear) time window.

5. Extensions: Random and Varying Environments

Critical finite variance branching extends naturally to models where the offspring law depends on an external (random or deterministic) environment:

• Varying environment: If offspring mean and variance vary across generations (BPVE), extinction, survival probabilities, and Yaglom-type exponential limits persist under corresponding criticality and variance summability conditions (Kersting, 2017).
• Branching in Markov/random environment: In random or Markovian environments, survival and population-size conditioned processes converge (after suitable normalization) to non-degenerate limits involving the random environment walk, e.g., $Z_n / e^{S_n}$, where $S_n = \sum_{k=1}^n \log f_{X_k}'(1)$ (Grama et al., 2024, Vatutin et al., 22 Jun 2025). The limit laws may involve Rayleigh-type distributions and explicit dependence on environmental stationarity.
• BPRE reduced processes: The number of ancestral particles at time $r$ surviving to time $n$ (the reduced process $Z_{r,n}$) connects to Brownian meander limits via associated random walks, under finite variance conditions (Vatutin et al., 22 Jun 2025).

These generalizations demonstrate the robustness of critical branching universality classes under moderate external fluctuations, provided variance-control is maintained.

6. Extreme Value Theory in Branching: Maximal Offspring and Heavy Tails

The study of extreme reproduction events within critical finite variance branching processes elucidates the interplay between overall population scaling and offspring distribution tails:

• Maximal offspring: For offspring laws with regularly varying tails of index $\alpha > 2$ (so finite variance), the maximal number of offspring amongst all individuals, suitably normalized, converges to a Fréchet law of shape $\alpha/2$ (Bertoin, 2012).
• Ranked large offspring: The vector of order statistics of large offspring exhibits Cox process convergence, with parameters depending on offspring variance and heavy-tail index; the finite variance regime sharply contrasts with the infinite (heavy tail) variance regime (Bertoin, 2012).

This highlights the dichotomy between stable Gaussian scaling of population-wide statistics and the possible atypical presence of large reproductive outliers in finite populations.

7. Technical Framework: Martingales, Functional CLT, and Proof Strategies

A common mathematical structure unifies the proofs and limit theorems across models:

• Spine techniques and martingale change of measure: Many-to-one formulas and spine decompositions play a central role in conditioning and limit law derivation (Horton et al., 9 Jan 2026, Powell, 2015).
• Depth-first exploration martingales: Height processes or Lukasewicz path encodings, when rescaled, converge to Brownian motion or excursions via functional central limit theorems, enabling identification of genealogical shape limits (Horton et al., 9 Jan 2026, Powell, 2015).
• Generating function and renewal methods: Classical and renewal approaches control critical populations, small-window reductions, and computation of ancestral structures via precise generating function expansions and Tauberian theorems (Liu et al., 2018, Hong et al., 2018).
• Gromov–Hausdorff–weak topology: The measured tree convergence to CRT uses Gromov–weak topology (via monomial functionals), upgraded by lower mass-bound arguments to Gromov–Hausdorff–weak topologies (Horton et al., 9 Jan 2026).

The technical apparatus relies crucially on second-moment control and martingale convergence, with universality of the exponential and Brownian limits dependent on the preservation of finite variance throughout branching and environmental mechanisms.

---

For further detailed statements, explicit formulas, and proof methodologies, see (Horton et al., 9 Jan 2026, Hong et al., 2018, Powell, 2015, Kersting, 2017, Ren et al., 2013, Vatutin et al., 22 Jun 2025, Grama et al., 2024, Liu et al., 2018, Bertoin, 2012), and (Sagitov, 2021).