Jagers–Nerman Theorem in Branching Processes

• The Jagers–Nerman theorem is a strong law of large numbers for supercritical Crump–Mode–Jagers processes, ensuring that normalized population counts converge almost surely to a nondegenerate limit.
• It employs a blend of martingale techniques and renewal theory to rigorously analyze asymptotic behavior and the impact of underlying reproduction measures.
• Recent extensions incorporate central limit theorems to characterize fluctuations around the mean, providing detailed insights into complex population dynamics.

The Jagers–Nerman theorem provides a law of large numbers and associated limit theorems for general (supercritical) branching processes of Crump–Mode–Jagers (CMJ) type, encompassing classical Galton–Watson processes and their continuous-time, age-structured, and multi-type extensions. Formulated in the framework of random-characteristic-counted population processes, its core result states that, under mild integrability and supercriticality conditions, the population—when suitably exponentially normalized—converges almost surely (a.s.) to a nondegenerate limit. The result is fundamental for the probabilistic analysis of population dynamics across general branching models and is the canonical strong law of large numbers for these systems. Extensions include second-order central limit theorems characterizing the nature of fluctuations about the mean, with the fluctuation scaling and limiting distribution depending sensitively on the underlying model class and additional moment/regularity conditions.

1. Model Setup: Crump–Mode–Jagers Branching Processes

CMJ branching processes describe populations where each individual lives for a random lifetime, reproduces according to a random point process (possibly with age- or individual-dependence), and is assigned an independent stochastic “characteristic” process. Formally, individuals $u$ in the Ulam–Harris labeling are equipped with independent copies $(\zeta_u,\,\xi_u,\,\varphi_u)$, where:

• $\zeta_u$ is the lifetime,
• $\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$ is the reproduction point process (children born at ages $X_{u,j}$),
• $\varphi_u(t)$ is a random characteristic.

Given these, the population process counted with characteristic $\varphi$ is

$Z_t^\varphi = \sum_{u \in I} \varphi_u(t - S(u)),$

where $S(u)$ is the birth time of individual $u$ ($(\zeta_u,\,\xi_u,\,\varphi_u)$0). The “plain” population size at time $(\zeta_u,\,\xi_u,\,\varphi_u)$1 is obtained for $(\zeta_u,\,\xi_u,\,\varphi_u)$2; cumulative birth count corresponds to $(\zeta_u,\,\xi_u,\,\varphi_u)$3 (Holmgren et al., 2016, Iksanov et al., 2021).

The mean reproduction measure is $(\zeta_u,\,\xi_u,\,\varphi_u)$4, and the process is supercritical if $(\zeta_u,\,\xi_u,\,\varphi_u)$5. There exists a unique Malthusian parameter $(\zeta_u,\,\xi_u,\,\varphi_u)$6 solving

$(\zeta_u,\,\xi_u,\,\varphi_u)$7

which determines the asymptotic exponential growth rate.

2. The Jagers–Nerman Law of Large Numbers

Under nonlattice and integrability conditions, and supercriticality (i.e., expected number of offspring per individual $(\zeta_u,\,\xi_u,\,\varphi_u)$8), the normalized population process converges almost surely and in $(\zeta_u,\,\xi_u,\,\varphi_u)$9. Precisely, if $\zeta_u$0, then

$\zeta_u$1

where $\zeta_u$2 is the terminal value of Nerman's martingale ($\zeta_u$3), nondegenerate on the event of nonextinction, and $\zeta_u$4 (Iksanov et al., 2020, Holmgren et al., 2016, Iksanov et al., 2021).

In the special case of the binary homogeneous Crump–Mode–Jagers process, the asymptotic limit has an explicit representation in terms of the scale function $\zeta_u$5 with Laplace transform $\zeta_u$6, with $\zeta_u$7 the associated Laplace exponent, and

$\zeta_u$8

almost surely and in $\zeta_u$9, with $\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$0 conditioned on survival (Henry, 2015).

3. Martingale Techniques and Proof Structure

The proof constructs the fundamental martingale

$\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$1

which is nonnegative and $\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$2-bounded under suitable integrability of the offspring and characteristic distributions. The proof utilizes:

• Branching process decomposition at the “coming generation,” exploiting the Markov and independence properties,
• Renewal theory to analyze mean behavior and the impact of the reproduction measure,
• The many-to-one lemma (change of measure) for first-moment estimates,
• Martingale convergence theorems to obtain $\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$3, $\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$4, and $\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$5 convergence of $\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$6,
• Uniform integrability to allow dominated convergence for general characteristics (Iksanov et al., 2020, Holmgren et al., 2016).

The direct Riemann integrability of $\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$7 ensures the requisite summability for applying renewal theorems.

4. Second-Order Limit Theorems: Central Limit Behaviour

Beyond the law of large numbers, recent developments have clarified the nature of stochastic fluctuations about the deterministic exponential growth:

• For the supercritical binary homogeneous CMJ model, the error $\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$8 satisfies

$\xi_u = \sum_{j=1}^{N_u} \delta_{X_{u,j}}$9

where the limiting law is double-exponential rather than Gaussian, and the scale $X_{u,j}$0 reflects the precise fluctuation magnitude (Henry, 2015).

• In general CMJ models with sufficient moment assumptions, there exist $X_{u,j}$1 and functions $X_{u,j}$2, so that

$X_{u,j}$3

converges in distribution to a normal variable with random variance, where the structure of $X_{u,j}$4 encodes subleading deterministic and principal martingale terms associated to additional roots of the Laplace exponent on the critical line $X_{u,j}$5 (Iksanov et al., 2021).

5. Functional Extensions and Gaussian Process Limits

In the regime of large populations (many initial ancestors), one obtains functional (process-level) laws of large numbers and central limit theorems in the canonical functional space $X_{u,j}$6:

• Under boundedness and regularity conditions on lifetimes, reproduction rates, and multiple-birth distributions, the time-indexed renormalized process

$X_{u,j}$7

satisfies

$X_{u,j}$8

pointwise and in $X_{u,j}$9, with $\varphi_u(t)$0 the solution to the mean renewal equation.

• The corresponding fluctuation process, scaled by $\varphi_u(t)$1, converges in $\varphi_u(t)$2 to a mean-zero Gaussian process with explicit, model-dependent covariance structure, decomposing into independent contributions from the randomness of lifetimes, birth rates, and Poissonian births, with technical tightness established via moment bounds for compensated Poisson integrals (Dramé et al., 16 Aug 2025).

6. Applications and Generalizations

The Jagers–Nerman theorem and its extensions underpin asymptotic analyses of a wide range of stochastic population models:

• Tree asymptotics for random structures such as $\varphi_u(t)$3-ary search trees, fringe trees, and random recursive trees, via embedding in CMJ frameworks (Holmgren et al., 2016).
• Multi-type branching and models with general random characteristics,
• Population genetics and mathematical biology, especially when structures cannot be represented by classical Galton–Watson models,
• The explicit nature of second-order expansions enables precise probabilistic control in high-dimensional stochastic systems.

The methodology has been further refined for lattice settings, random age-dependent birth rates, and characteristic processes with nontrivial time-dependence or sign variation.

7. Selected Formulas and Key Quantities

Quantity	Formula	Reference
Malthusian parameter $\varphi_u(t)$4	$\varphi_u(t)$5	(Iksanov et al., 2021, Holmgren et al., 2016)
Characteristic mean	$\varphi_u(t)$6	(Holmgren et al., 2016)
Normalized population	$\varphi_u(t)$7 a.s.	(Iksanov et al., 2021, Iksanov et al., 2020)
Binary splitting CLT scale	$\varphi_u(t)$8	(Henry, 2015)
Gaussian process covariance	$\varphi_u(t)$9 (explicit expression)	(Dramé et al., 16 Aug 2025)

These expressions codify the structural outputs of the Jagers–Nerman theorem, with detailed covariance formulas and Laplace transforms providing concrete analytical tools for evaluating large-time asymptotics and fluctuation behaviour.

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References:

• (Henry, 2015) Central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processes
• (Iksanov et al., 2020) Gaussian fluctuations and a law of the iterated logarithm for Nerman's martingale in the supercritical general branching process
• (Holmgren et al., 2016) Fringe trees, Crump-Mode-Jagers branching processes and $\varphi$0-ary search trees
• (Iksanov et al., 2021) Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes
• (Dramé et al., 16 Aug 2025) Functional law of large numbers and central limit theorem for Crump-Mode-Jagers branching processes