Absolute continuity of the martingale limit in branching processes in random environment

We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $\xi =(\xi_n)_{n\ge0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_n=Z_n/ (\mathbb E (Z_n|\xi ))$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $\xi$ the law of $W$ conditioned on the environment $\xi $ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of \cite{kaplan:1974}, and of course it covers the well-known case of the martingale limit of a Galton-Watson process. Our proof combines analytical arguments with the recursive description of $W$.


Introduction and statement of the main result
We consider a supercritical branching process Z n in a stationary and ergodic random environment ξ = (ξ n ) n≥0 , defined as follows. Let ∆ be the space of probability measures on N 0 = {0, 1, 2, ...} -the set of possible offspring distributions. Let ξ = (ξ n ) n≥0 be a stationary and ergodic process taking values in ∆. The sequence (ξ n ) n≥0 is called a "random environment" or "environment sequence". All our random variables are defined on a probability space (Ω, F, P). The process (Z n : n ≥ 0) with values in N 0 is called a branching process in random environment ξ if Z 0 is independent of ξ and it satisfies L(Z n |ξ, Z 0 , . . . Z n−1 ) = ξ * Zn−1 n−1 a.s. (1.1) where ξ * k n−1 is the k fold convolution. Conditioned on the past and on the environment sequence, Z n may be viewed as the sum of Z n−1 independent and identically distributed random variables Y n−1,i , each having law ξ n−1 . The process {Z n } ∞ By P ξ we denote the measure P conditioned on the environment ξ. The corresponding mean and variance are denoted by E ξ and Var ξ i.e. for any random variable X we have E ξ [X] = E[X|ξ] and Var ξ (X) = E (X − E ξ X) 2 |ξ . Finally, for any random variable X we also introduce the conditional law L ξ (X) by L ξ (X)(A) = P(X ∈ A|ξ), where the equality is valid for any measurable set A. Then (W n ) n≥0 is a nonnegative martingale under P ξ . Therefore, lim n→∞ W n = W exists P ξ -almost surely. There has been a lot of interest in asymptotic properties of W , convergence rates of W − W n as well as limit theorems for Z n and large deviations principles. Positive and negative, annealed and quenched, moments of W were investigated. Most of that was done for i.i.d. environments, because then properties of the so-called "associated random walks" could be applied, but some results hold also in a stationary and ergodic environment. For a sample of results see [3,5,4,8,9] and references therein.
However, except of [10] the local regularity of the law of W has not been studied. Due to the recursive equation (1.2) satisfied by W , see below, it is closely related to the local regularity for stationary solutions to affine type equations, see (1.3) below. This motivated us to study absolute continuity of the law of W . More precisely, the following recursive formula will be crucial for our proof. The definition of the process Z n yields that W satisfies the relation where under P ξ , the random variables W j are independent of each other and independent of Z 1 with distribution P ξ (W j ∈ ·) = P T ξ (W ∈ ·). Here, T is the translation operator defined by T (ξ 0 , ξ 1 , . . . ) = (ξ 1 , ξ 2 , . . . ). The question about local regularity of L ξ (W ) fits very well into a number of similar problems being investigated recently [6,7,12,13,18,22]. For the Galton Watson process the W j 's have the same law as W and so then (1.2) is an example of the so-called smoothing equation. By the latter we mean where the equality is meant in law, (C, A 1 , A 2 , . . . ) is a given sequence of real or complex random variables and Y 1 , Y 2 , . . . are independent copies of the variable Y and independent of (C, A 1 , A 2 , . . . ). Let N be a random number of A j 's that are not zero. As long as EN > 1 the transform where µ is the law of Y 1 , improves local regularity of the measure, and so it is expected that the fixed points of S are absolutely continuous even when the A j 's and C are discrete. This is indeed the case, see [7], [12], [13] and references therein. However, in the case of a random environment, the equation (1.2) is not exactly of the form in (1.3) and so a different approach had to be elaborated.
and absolute continuity of the solution is much harder to prove if (A, C) does not possess a priori any regularity, as for instance in the case of Bernoulli convolutions A is concentrated at λ, for some 0 < λ < 1 and C is a Bernoulli random variable, i.e. C takes the values +1, −1 each with probability 1/2. If 0 < λ < 1/2 then the law ν λ of Y = λY + C is continuous but singular with respect to Lebesgue measure and if λ = 1/2 then ν λ is the uniform distribution on [−2, 2]. However, when 1/2 < λ < 1, ν λ is absolutely continuous for almost every such λ or even better: it is absolutely continuous outside of a subset of λ ∈ (1/2, 1) of Hausdorff dimension 0. Moreover, if particular λ's are considered, absolute continuity of ν λ depends on delicate algebraic properties of λ, see [22] for an overview of the recent developments on Bernoulli convolutions. When we go beyond Bernoulli convolutions there is no general theory about regularity of ν. Further examples of singular (A, C) that give rise to absolutely continuous solutions as well as to singular ones are available, see [6], [14], [18]. Let q(ξ) = P ξ lim n→∞ Z n = 0 Z 0 = 1 be the extinction probability of the process Z n . Since ξ is ergodic, P(q(ξ) < 1) equals 0 or 1 a.s.. We assume that the random variable log m 0 is integrable. If E log m 0 ≤ 0 then it is easy to see that P(q(ξ) = 1) = 1, see also [19], unless ξ 0 = δ 1 a.s. Therefore, we will The question whether P(q(ξ) < 1) is 0 or 1 is well understood (c.f. [16,17,2] and [11]): holds. Then P(q(ξ) < 1) = 1.
Our main result is the following description of the law of W under P ξ . (ii) L ξ = δ 1 a.s. In order to prove Theorem 1.2, we will need some additional statements provided in the next section.

Further results
In general, for a supercritical BPRE, W may vanish almost surely and conditions for that to happen are well known. Recall that L ξ is the law of W under P ξ . Notice that due to (1.2), the sets {ξ : L ξ = δ 0 } and {ξ : L T ξ = δ 0 } coincide. Therefore, by ergodicity, P(ξ : In fact, as explained below, it is known, that if z(ξ) < 1 then z(ξ) = q(ξ) but we will not need this information for our proof of Theorem 2.1. We say that a measure is degenerate if it is concentrated at a point. Theorem 2.1. Suppose that the environment sequence ξ is stationary and ergodic, (1.5) holds, P(ξ : L ξ = δ 0 ) = 0 and P(ξ : ξ 0 not degenerate) > 0. Then where ν ξ is absolutely continuous with respect to Lebesgue measure. Remark 2.2. Theorem 1.2 follows directly from Theorem 2.1. Indeed, if P(ξ : L ξ = δ 0 ) = 1 then (i) in Theorem 1.2 holds. If µ > 0 (recall (1.5)) and P(ξ : ξ 0 degenerate) = 1 then W n is concentrated at 1 for every n, hence the same is true for W . Moreover, if W is not identically zero then z(ξ) = q(ξ), see [20] and [21]. Let us provide a short argument for the latter statement. If (1.5) holds, then by [20,Theorem 1] there exists a sequence of random variables c n (ξ) and a nonnegative random variable U such that On the other hand, L(ξ) is constant under P ξ , and therefore, L(ξ) = 0 would imply P ξ (W = 0) = 1 which is a contradiction. Hence W = L(ξ)U and P ξ (W = 0) = P ξ (U = 0) = q(ξ).

(2.4)
Moreover, it was proved in [21] that if (ξ n ) is an i.i.d. sequence then condition (2.2) is in fact equivalent to (2.3). Another proof for i.i.d. environments (ξ n ) is contained in [11]. For i.i.d. environments, assuming (1.5), (2.2) and E ξ W = 1 a.s. are equivalent. In general, when the sequence (ξ n ) is assumed to be only stationary and ergodic (2.2) is not necessary for W to be not identically zero [21]. In this case the necessary condition is ∞ n=0 m −1 n k≥Mn+1 kξ n (k) < ∞ a.s. The sufficient condition is only a little bit stronger (see Theorem 1, [21]). Under this sufficient condition, (2.4) holds.
We write for the conditional characteristic function of W . We now derive a second recursive formula which is crucial for our proofs. Define Then by the recursive relation (1.2) we obtain ψ(t, ξ) = f 0 (ψ(t/m −1 0 , T ξ)) = f 0 • · · · • f n−1 (ψ(t/M n , T n ξ)) = F n (ψ(t/M n , T n ξ)), (2.6) where F n is the probability generating function of Z n given by (2.5) and T the translation operator defined above.
In order to prove Theorem 2.1 we use the following analytical result.
Lemma 2.4. Let ν be a probability measure on (R, B) with finite first moment and let ψ be its characteristic function. If |ψ | is integrable then ν = cδ 0 + ν abs where ν abs is absolutely continuous with respect to the Lebesgue measure.
Proof. ∂ t ψ(t) dt defines a tempered distribution, see [15], part 2. Moreover, its Fourier inverse satisfies where f is a complex valued function vanishing at infinity. In the above formula the first F −1 means the inverse Fourier transform of a tempered distribution and the second F −1 the inverse Fourier transform of an integrable function. On the other hand This shows that ν1 R\{0} has density given by ix −1 f (x) and the conclusion follows.
ECP 24 (2019), paper 42. Remark 2.5. Theorem 2.1 generalizes considerably Theorem 1 in [10] but, what is more important, Kaplan's proof contains essential gaps that concern the integrability of |ψ (·, ξ)|. We don't think that they are easily reparable within his approach and instead we suggest our proof which is contained in Theorem 2.6 below. However, the idea to show the integrability of |ψ (·, ξ)| is borrowed from [10].
The key step in the proof of Theorem 2.1 is the following theorem.
It turns out that (2.7) can be quite easily guaranteed.

Proof of Theorem 2.7
We first need some auxiliary results. Lemma 3.1. Suppose that W is not identically zero and W is degenerate, i.e. Var ξ W = 0. Then P(ξ 0 is degenerate) = 1.
Proof. Taking conditional expectation of both sides of (1.2), we see that E ξ W = E T ξ W and so by ergodicity, E ξ W is a strictly positive constant, call it γ. Moreover, due to (1.2), (which holds also in the case when one of the terms is infinite). Suppose that Var ξ W = 0. Then iterating (3.1), we have that Var T i ξ Z 1 = 0 for all i ∈ N, which is not possible. Indeed, if P(ξ 0 is not degenerate) > 0 then by Birkhoff's ergodic theorem for a.e. ξ there is i such that (T i ξ) 0 = ξ i is not degenerate.

Lemma 3.2.
Assume that W is not identically zero and that P(ξ 0 not degenerate) > 0 and (1.5) holds. Then there is a measurable function ξ → (N (ξ), c(ξ)) ∈ N × [0, 1] such that for a.e. ξ, c(ξ) > 0 and Proof. Let W be a random variable such that under P ξ , W and W are i.i.d. Then for almost all ξ we have by the monotone convergence theorem. It follows that, for c(ξ) : on some neighbourhood of 0. In particular, and since ξ → ψ(t, ξ) is measurable, τ is measurable as well. The lemma now holds with N (ξ) := τ (ξ) −1 .

Lemma 3.3.
Assume that the environment sequence ξ is stationary and ergodic such that (1.5) holds. If W is not degenerate then for any 0 < β < 1 there are constants c > 0 and t 0 ≤ 1 such that for a.e. ξ there is a sequence of natural numbers n i such that and |ψ(t, T ni ξ)| ≤ 1 − ct 2 , f or 0 ≤ t ≤ t 0 .
Proof of Theorem 2.7. We write, using (2.6), ECP 24 (2019), paper 42. and the absolute value of the first term above is bounded by P ξ (W = 0) < 1. It remains to show that the second term converges to zero as |t| → ∞ (n = n(t) will be adjusted to t). Fix 0 < β < 1 25 . Then, by the ergodic theorem we get that for almost every ξ for sufficiently large i. In view of (3.4) and (3.3), for large i we have Moreover, we may assume that i is maximal with that property. Then x < t 0 M ni+1 and Since, for large enough i, we have M ni+1 ≥ e (1−β) 2 (i+1)µ and by the choice of β, (1−β) 2 > 6β, the dominated convergence theorem gives

Integrability of ψ
In this section we prove Theorem 2.6. To this end, we need the following auxiliary result.
The idea to consider the function h is borrowed from [5].
for some 1 − r < s < 1 and since all the derivatives are positive we can take the limit for N → ∞ and obtain the desired inequality. Next, we conclude that where the reminder R 1 is given by From the fact that ξ 0 (0) < 1 a.s. we infer Since all derivatives of f are nonnegative and so nondecreasing, we conclude In particular, we can estimate the reminder from below Now we are ready to prove Theorem 2.6, but first let us sketch the idea. Similarly to the proof of Theorem 2.7 the intervals [M ni , M ni α i ] cover some half line [y, ∞) and therefore the integrability of ψ will follow once we prove the finiteness of i |t|∈Ii |ψ (t)|dt, On each such interval I i we can use the relation (2.6) and then apply the chain rule. By doing so we get a product of derivatives of functions f i which in general is not easy to handle. However, replacing f i by h i , which are bounded by 1 and counting those that are bounded away from 1 leads to exponential decay of |ψ (t)|, uniformly for M ni ≤ |t| ≤ M ni α i . Finally, as α i = e 3βi with arbitrary small β we conclude the integrability of ψ .