Convergence of complex martingale for a branching random walk in a time random environment

We consider a discrete-time branching random walk in a stationary and ergodic environment ξ = ( ξ n ) indexed by time n ∈ N . Let W n ( z ) ( z ∈ C d ) be the natural complex martingale of the process. We show sufﬁcient conditions for its almost sure and quenched L α convergence, as well as the existence of quenched moments and weighted moments of its limit, and also describe the exponential convergence rate.


Introduction and main results
We consider a branching random walk in a time random environment (BRWRE), where the distributions of the point processes indexed by particles vary from generation to generation according to a time random environment. First introduced by Biggins and Kyprianou [4], this model was further studied in [9,11,21,23]. For the classical branching walk, Biggins [3] showed a sufficient condition for the almost sure and L α convergence of the complex martingale of the model for α ∈ (1,2], and recently, necessary and sufficient conditions for α > 1 were shown by Ikzanove et al. [13], while Kolesko and Meiners [15] especially discussed the convergence on the boundary of the uniform convergence region. Aiming to extend the result of [3], this paper focuses on investigating the convergence (in the sense almost sure and in L α for α > 1) of the complex martingale in BRWRE. The main results presented in the paper cannot be derived directly by techniques suitable for classical branching walks. The main reason is that the environment makes it difficult to find useful upper bounds of martingales. Similar problems may appear in other models in random environments, such as branching processes, multiplicative cascades and random fractals in random environments, etc. The techniques used in the paper, especially in the side of dealing with the stationary and ergodic random environments, should provide reference for related topics.
Let us describe the model in detail. The time random environment, denoted by ξ = (ξ n ), is a stationary and ergodic sequence of random variables, indexed by the time n ∈ N = {0, 1, 2, · · · }, taking values in some measurable space (Θ, E). Without loss of generality we can suppose that ξ is defined on the product space (Θ N , E ⊗N , ν), with ν the law of ξ. Stationarity means that the two random vectors (ξ k , ξ k+1 , · · · , ξ k+n ) and (ξ k+h , ξ k+1+h , · · · , ξ k+n+h ) have the same joint distribution for any k, n and h ∈ N; ergodicity can be comprehended as that the following Birkhoff ergodic theorem holds: for any measure-preserving transformation τ and integrable function f on (Θ N , E ⊗N , ν), for almost all ξ.
For each realization of ξ n , there exists a distribution on N × (R d ) ⊗N * corresponds it, where d ≥ 1 is the dimension of the real space and N * = {1, 2, · · · }. We denote the distribution corresponding to ξ n by η n = η(ξ n ). The notation η(ξ n ) can be regarded as a mapping from the space (Θ, E) to the set of all distributions on N × (R d ) ⊗N * . Given the environment ξ, the process can be described as follows: at time 0, one initial particle ∅ of generation 0 is located at S ∅ = 0 ∈ R d ; in general, each particle u of generation n located at S u ∈ R d is replaced at time n + 1 by N (u) new particles ui of generation n + 1, where the random vector X(u) = (N (u), L 1 (u), L 2 (u), · · · ) is of distribution η n = η(ξ n ); all particles behave independently conditioned on the environment ξ.
For each realization ξ of the environment sequence, let (Γ, G, P ξ ) be the probability space on which the process is defined. The probability P ξ is usually called quenched law, while the total probability P is usually called annealed law. The quenched law P ξ may be considered to be the conditional probability of P given ξ. The expectation with respect to P (resp. P ξ ) will be denoted by E (resp. E ξ ).
Let U = {∅} ∪ n≥1 (N * ) n be the set of all finite sequence u = u 1 · · · u n . For u ∈ U, we write |u| for the length of u. Let T be the Galton-Watson tree with defining elements {N (u)} and T n = {u ∈ T : |u| = n} be the set of particles of generation n. For n ∈ N and z = x + iy ∈ C d , put e zLi(u) (|u| = n), (1.1) where the product zL should be understood as the inner product that We consider the non trivial case that P ξ (N = 0) < 1 a.s., (1.2) so that m n (z) = 0 a.s. Set For z ∈ C d and u ∈ T, denote X u (z) = e zSu P |u| (z) , (1.4) Let F 0 = σ(ξ) and F n = σ(ξ, X(u); |u| < n) for n ≥ 1. It is well known that for each z ∈ C d fixed, W n (z) forms a complex martingale with respect to the filtration F n under both laws P ξ and P. Particularly, for x ∈ R d , the martingale W n (x) is non-negative, hence it converges almost surely (a.s.). In the deterministic environment case, this martingale (with real or complex parameters) has been studied by Kahane and Peyrière [14], Biggins [2,3], Uchiyama [22], Durrett and Liggett [7], Guivarc'h [10], Lyons [20] and Liu [18,19], etc. in different contexts. In this paper, we are interested in the convergence of the complex martingale W n (z) for z ∈ C d fixed. For simplicity, later we write X u = X u (z) and W n = W n (z) for short.
In deterministic environment, Biggins ([3], Theorem 1) showed a sufficient condition for the almost sure and L α convergence of W n for α ∈ (1, 2], but there was no information for the case α > 2. When the environment is independent and identically distributed (i.i.d.), we can deduce the following result from ( [11], Theorem 2.4) without effort, which completes and generalizes the results of [3,11].
if the expectation exists as real number.
However, when the environment is stationary and ergodic rather than i.i.d, there were no corresponding results in the literature. Many times the methods available for i.i.d environments could not be applied directly to stationary and ergodic environments. For our problem, the main trouble is that it is difficult to estimate the upper bounds Similar trouble was also encountered during our study on the L α convergence rate of the real martingale in [23], where we obtained satisfactory result for the i.i.d environment case, but failed to acquire the corresponding result for all α > 1 in the stationary and ergodic environment case. Such difficulty has been overcome in this paper. Instead of finding the direct upper bounds, we have discovered the asymptotic upper bounds (see Theorem 2.3), with which we successfully obtain the corresponding results of Theorem 1.1 for the stationary and ergodic environment case.
For z = x + iy ∈ C d fixed, write f z (s) = E log m 0 (sx) − sE log |m 0 (z)| (s ∈ R) if the expectations exist as real numbers.
s., so that W n converges a.s. and in P ξ -L α for almost all ξ. Remark 1.3. (a) Apparently, the long-term behaviors of branching random walks can be investigated with the help of the additive martingale W n . For example, we can use Theorem 1.2 to give a sharp upper bound for the deviation Z n+1 (z)−m n (z)Z n (z), where Z n (z) := u∈Tn e zSu . Besides, in the study of the asymptotic behaviors of BRWRE, it is often necessary to check the convergence of the series in the form of n a 0 · · · a n−1 E T n ξ (W * ) α , where a n is a random variable depending on ξ n and T is the shift operator satisfying T n ξ = (ξ n , ξ n+1 , · · · ) if ξ = (ξ 0 , ξ 1 , · · · ). In this case, we need to first ensure the finiteness of the moment E ξ (W * ) α before going a step further. It is also worth mentioning that the method presented in this paper may provide an available approach for the study of the convergence of the series mentioned above.
(b) From Theorem 1.2, we can see that W n converges a.s. and in P ξ -L α for almost all ξ to a non-trivial limit (pointwisely) on the set In deterministic environment, Kolesko and Meiners [15] studied the convergence of W n on the boundary of Λ. Their method can be extended to work on the analogous boundary problem for BRWRE with i.i.d. environment. However, for the stationary and ergodic environment case, as the boundary condition cannot ensure the trueness of the so-called many-to-one formula, the convergence of W n on the boundary of Λ is still an open question.
In order to help readers better understand the set Λ in BRWRE and distinguish it from the one in classical branching random walk, we present below a simple example corresponding to Example 3.1 of [15].
Example 1.4 (Binary splitting with Gaussian increments in a random environment). Given the environment ξ = (ξ n ), we consider a branching random walk on R with independent Gaussian increments and binary splitting, i.e., X(u) = (2, It is not hard to detect that the shape of Λ is similar to ([15], Figure 1) but with some minor changes in coordinates. Particularly, in the case where E(1/σ 2 0 ) = 1, the figure of Λ coincides with ([15], Figure 1).
Under stronger conditions, we can further obtain the existence of the quenched In i.i.d. environment, corresponding annealed weighted moments can be deduced from Liang and Liu ([17], Theorem 1.1).
Moreover, thanks to Theorem 2.3, we can further investigate the exponential rate of the quenched L α convergence of W n to its limit, denoted by W if it exists. Theorem 1.6 (Quenched L α convergence rate). Let α > 1 and ρ > 1.
For R-valued BRWRE (i.e. the space dimension d = 1), Wang and Huang ([23], Theorem 1.1) showed the exponential rate of the quenched L α convergence of the non-negative martingale W n (x) for 1 < α ≤ α x , where α x ∈ (0, ∞] depending on x is a general constant that can be calculated accurately. The evident pity in that result is the lack of the description for the case α > α x . Theorem 1.6 remedies this lack, and meanwhile generalizes the result to the complex martingale W n (z) in R d -valued BRWRE.
s. Furthermore, we can even find the asymptotic upper bounds for E ξ (Y

Proof of Theorems 1.2-1.6
Let us come back to BRWRE and give the proofs of Theorems 1.2-1.6. As Theorem 1.1 can be proved by arguments in the proof of ( [11], Theorem 2.4) with tiny modifications, we omit its proof.
For z = x + iy ∈ C d fixed, setX u = |X u | = e xSu |P |u| (z)| . We will use the notations introduced in Section 2 for the Mandelbrot martingale. Note that nowΛ(s) = f z (s). Moreover, for u ∈ U, denote It can be seen that if E log + E ξ ( u∈T1X u ) α < ∞, then E log + ζ (s) 0 < ∞ for 0 < s ≤ α. In the following proofs, we will use the generalized BDG-inequality for complex martingales (we still call it the BDG-inequality later). Such inequality can be obtained by applying the classical BDG-inequality for real-valued martingales to the real and imaginary parts respectively of the complex martingales and noticing the convexity and monotonicity of the related functions.