Biggins Martingale in Branching Random Walks

• Biggins martingale is the additive martingale for supercritical branching random walks, defined via exponential tilting and Laplace normalization.
• It extends to complex parameters, underpinning techniques to study growth, change of measure, and extreme-value behavior in branching processes.
• Key results address convergence, derivative martingale behavior, and fluctuations, with implications for Lévy and matrix generalizations.

The Biggins martingale is the additive martingale associated with a supercritical branching random walk, defined by an exponential tilt of the generation-$n$ point process and a normalization by its Laplace transform. In the real-parameter setting it is typically written

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$

while for a complex parameter $\lambda$ it takes the form

$$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$

It is a central object in the study of branching random walks, where it governs growth, change of measure, extremal behavior, and stochastic fixed-point equations; later work extends the same structure to complex parameters, branching Lévy processes, and matrix branching random walks (Aidekon et al., 2011, Iksanov et al., 2019, Bertoin et al., 2017, Grama et al., 13 Jul 2025).

1. Definition and basic framework

A branching random walk on $\mathbb R$ starts with one ancestor at the origin. Each particle produces offspring whose locations relative to the parent are given by a point process, and the offspring then reproduce independently according to the same law. If $V(x)$ denotes the position of a particle $x$ in generation $|x|=n$, then the additive martingale at parameter $t$ is

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$

When the critical parameter is normalized to $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$0 and $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$1, this reduces to

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$2

Conditioning on the first $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$3 generations shows that $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$4 is a martingale with respect to the natural filtration, and for $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$5 one has $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$6 (Aidekon et al., 2011).

The same construction admits a complex extension. For $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$7 with $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$8,

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$9

is a complex-valued martingale with mean $\lambda$0. The real and complex theories share the same branching structure, but the complex case loses positivity and therefore requires substantially finer moment and spectral arguments (Iksanov et al., 2019).

The martingale can also be embedded in a weighted branching process. Writing weights $\lambda$1 on offspring and

$\lambda$2

one obtains the generation sum

$\lambda$3

For branching random walks with complex parameter $\lambda$4, the identification

$\lambda$5

gives

$\lambda$6

so Biggins’ martingale is a particular instance of a weighted branching martingale and of the smoothing-transform framework (Damek et al., 2018).

2. Convergence, non-degeneracy, and $\lambda$7 behavior

For real parameters, classical convergence questions ask whether the additive martingale converges almost surely, whether the limit is nontrivial, and under which conditions convergence holds in $\lambda$8. In the real-parameter case recovered in the complex-parameter study, $\lambda$9 converges in $$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$0 for $$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$1 if and only if

$$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$2

which is the standard Biggins-type criterion for the nonnegative martingale (Iksanov et al., 2019).

For genuinely complex parameters, the picture splits according to whether $$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$3 or $$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$4. In the first case, the martingale reduces essentially to the real one. In the second, necessary and sufficient conditions for $$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$5-convergence are more involved. For $$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$6, one obtains a criterion in terms of the first-step moment $$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$7, a further condition $$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$8 when $$W_n(\lambda)=m(\lambda)^{-n}\sum_{|u|=n}e^{-\lambda S(u)}, \qquad m(\lambda)=E\Big[\sum_{|u|=1}e^{-\lambda S(u)}\Big].$$9, and a spectral inequality involving $\mathbb R$0, $\mathbb R$1, and $\mathbb R$2 (Iksanov et al., 2019). For $\mathbb R$3, the theory requires either a second-moment assumption or a regularly varying tail assumption for $\mathbb R$4, together with a Seneta–Heyde-type analysis of the corresponding real martingale (Iksanov et al., 2019).

In continuous time, branching Lévy processes provide the natural analogue of branching random walks. Their Biggins martingale is

$\mathbb R$5

where $\mathbb R$6 is the cumulant generating function expressed through the branching Lévy triplet $\mathbb R$7. A version of Biggins’ theorem holds in this setting: $\mathbb R$8 is uniformly integrable if and only if

$\mathbb R$9

and

$V(x)$0

otherwise $V(x)$1 almost surely (Bertoin et al., 2017). For $V(x)$2, later work gives ultimate necessary and sufficient conditions for $V(x)$3-convergence in terms of

$V(x)$4

and an explicit integrability condition on the offspring Lévy measure (Iksanov et al., 2018).

A recurrent misunderstanding is to regard nontrivial martingale convergence as generic. The literature instead separates a supercritical $V(x)$5-convergent regime, a critical boundary regime where the additive martingale degenerates, and heavy-tail or complex regimes where different normalizations become necessary (Bertoin et al., 2017, Aidekon et al., 2011).

3. Boundary case and derivative martingale

The most delicate real-parameter regime is the boundary case in the sense of Biggins–Kyprianou. After normalization to the critical parameter $V(x)$6, it is characterized by

$V(x)$7

together with

$V(x)$8

In this regime the additive martingale

$V(x)$9

still converges almost surely, but the limit is degenerate: $x$0 on the survival event (Aidekon et al., 2011, Madaule, 2016).

The nontrivial object is the derivative martingale

$x$1

Biggins and Kyprianou proved that under the boundary assumptions and logarithmic moment conditions,

$x$2

where $x$3 is finite and strictly positive on nonextinction (Aidekon et al., 2011, Madaule, 2016). In this setting the derivative martingale is literally the derivative, up to sign, of the additive martingale with respect to the parameter at the critical point.

Chen identified a sharp necessary and sufficient condition for the derivative martingale limit to be nontrivial. With

$x$4

the limit $x$5 is nontrivial if and only if

$x$6

This is a Kesten–Stigum-like criterion for the boundary derivative martingale and pins down exactly when the critical Mandelbrot cascade does not collapse (Chen, 2014).

A second major result is the Seneta–Heyde scaling. Under the boundary assumptions and the moment conditions

$x$7

with

$x$8

Aïdékon and Shi proved

$x$9

The convergence is in probability under the survival-conditioned law $|x|=n$0, not almost surely; in fact,

$|x|=n$1

This separates the additive and derivative martingales sharply: the additive martingale dies out, but at rate $|x|=n$2 it is asymptotically proportional to the derivative martingale (Aidekon et al., 2011).

The derivative martingale also governs extreme-value asymptotics. Its tail satisfies

$|x|=n$3

and the proof relates $|x|=n$4 to the global minimum $|x|=n$5 of the branching random walk through the factorization

$|x|=n$6

This connects large derivative-martingale values to very deep minima of the branching random walk (Madaule, 2016).

4. Complex parameters, smoothing transforms, and absolute continuity

For complex $|x|=n$7, Biggins’ martingale is

$|x|=n$8

In the weighted branching formulation, the corresponding limit $|x|=n$9 solves the complex smoothing equation

$t$0

where $t$1 are i.i.d. copies of $t$2, independent of the weights. The characteristic function then satisfies

$t$3

This places the limit of Biggins’ martingale within the general theory of stochastic fixed points and operator-stable decompositions (Damek et al., 2018).

A structural theorem recalled in the same setting states that, under assumptions $t$4–$t$5 and technical hypotheses with $t$6, any fixed point law can be written

$t$7

where $t$8 is the nonnegative martingale limit associated with $t$9, $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$0 is the martingale limit $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$1, and $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$2 is a Lévy component with operator-stable invariance. In finite-variance branching-random-walk applications the Lévy part vanishes, leaving $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$3; in that sense the limit of Biggins’ martingale is the martingale-type solution of the smoothing transform (Damek et al., 2018).

Damek and Mentemeier gave a simple criterion for the absolute continuity of the limit law. In the genuinely complex case, assume: $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$4 together with

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$5

Then the law of $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$6 is absolutely continuous on $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$7. If $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$8, the condition $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n}e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big].$$9 suffices in place of the stronger complex assumption (Damek et al., 2018).

For branching random walks, the specialization $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$00 identifies this result directly with the limit $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$01 of Biggins’ martingale. The paper emphasizes that “for given values of $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$02, the assumptions of Theorem $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$03 are readily checked,” and in the symmetric binary example computes explicitly

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$04

Thus, under suitable spectral and logarithmic moment conditions, the limit law of Biggins’ martingale with complex parameter has a density with respect to Lebesgue measure on $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$05 (Damek et al., 2018).

5. Fluctuation theory

Beyond convergence, a substantial literature analyzes the rate and mode of convergence of Biggins’ martingale to its limit. In the square-integrable regime, Iksanov and Kabluchko proved a functional central limit theorem for the tail process. Under the normalization $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$06, the assumptions

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$07

imply

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$08

where

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$09

and $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$10 is a stationary centered Gaussian sequence with covariance

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$11

They also proved a law of the iterated logarithm: $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$12 with the corresponding negative liminf (Iksanov et al., 2015).

In a heavy-tailed regime, the fluctuations cease to be Gaussian. If $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$13 belongs to the domain of normal attraction of an $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$14-stable law, with

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$15

and

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$16

in the notation of the paper, then the normalized tail process satisfies

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$17

where $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$18 is a stationary autoregressive process of order one with $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$19-stable marginals. This gives a stable analogue of the Gaussian tail-process limit theorem (Iksanov et al., 2017).

For complex parameters, the fluctuation picture is richer still. One study identifies three different regimes. First, for parameters with small absolute values, fluctuations are Gaussian and the limit laws are scale mixtures of the real or complex standard normal laws. Second, there is a region of parameter space in which the fluctuations are determined by the extremal positions in the branching random walk. Third, on a critical region, typically on the boundary of the set of parameters for which the martingales converge to non-degenerate limits, the fluctuations are stable-like and the limit laws are those of randomly stopped Lévy processes satisfying invariance properties similar to stability (Iksanov et al., 2018).

The same work also makes explicit that in the extremal regime the relevant normalization involves the minimal-position scale, while on the stable boundary $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$20 the fluctuations are described by a Lévy process $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$21 evaluated at a random time proportional to the derivative-martingale limit $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$22. This links additive-martingale fluctuations to both extremal process theory and smoothing-transform tail theory (Iksanov et al., 2018).

6. Continuous-time and higher-dimensional generalizations

The continuous-time analogue of Biggins’ martingale arises in branching Lévy processes. Such a process is determined by a characteristic triplet $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$23, and the additive martingale is

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$24

A branching-Lévy version of Biggins’ theorem gives necessary and sufficient conditions for a nondegenerate $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$25 limit, and an $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$26-criterion for $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$27 is later derived through Lévy-type perpetuities (Bertoin et al., 2017, Iksanov et al., 2018). A 2025 study continues this line by analyzing moment properties of $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$28 and $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$29, the tail behavior of $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$30, and central limit theorems for $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$31 (Ren et al., 11 Sep 2025).

The same general program extends beyond scalar spatial positions. In a matrix branching random walk on the semigroup of nonnegative matrices, the analogue of Biggins’ additive martingale is

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$32

where $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$33 is the positive eigenfunction of the transfer operator $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$34 and $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$35 is the logarithmic cocycle of the matrix product along the branch. Under general assumptions, a full analogue of Biggins’ martingale convergence theorem holds: for fixed $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$36,

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$37

The matrix theory also has a derivative martingale

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$38

which converges to $$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$39, and a Seneta–Heyde scaling

$$W_n(t)=e^{-n\psi(t)}\sum_{|x|=n} e^{-tV(x)}, \qquad \psi(t)=\log E\Big[\sum_{|x|=1}e^{-tV(x)}\Big],$$40

This generalization replaces scalar spatial homogeneity by spectral theory for products of random nonnegative matrices and a centered Markov random walk on projective space (Grama et al., 13 Jul 2025).

A plausible implication is that Biggins’ martingale should be viewed less as a single formula than as a template: a normalized multiplicative martingale built from a branching system and a spectral transform. In scalar branching random walks, complex weighted branching processes, branching Lévy processes, and matrix branching random walks alike, the decisive structures are the same: a Laplace or transfer operator, a martingale change of measure, a boundary regime with vanishing additive martingale, and a derivative martingale that captures the critical first-order mass (Damek et al., 2018, Bertoin et al., 2017, Grama et al., 13 Jul 2025).