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Fixed points of the multivariate smoothing transform: the critical case

Konrad Kolesko, Sebastian Mentemeier

2015
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Electronic Journal of Probability
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E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. Abstract Given a sequence (T1, T2, . . . ) of random d × d matrices with nonnegative entries, suppose there is a random vector X with nonnegative entries, such that i≥1 TiXi has the same law as X, where (X1, X2, . . . ) are i.i.d. copies of X, independent of (T1, T2, . . . ). Then (the law of) X is called a fixed point of the multivariate smoothing transform. Similar to the well-studied one-dimensional case d = 1, a function
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... d = 1, a function m is introduced, such that the existence of α ∈ (0, 1] with m(α) = 1 and m (α) ≤ 0 guarantees the existence of nontrivial fixed points. We prove the uniqueness of fixed points in the critical case m (α) = 0 and describe their tail behavior. This complements recent results for the non-critical multivariate case. Moreover, we introduce the multivariate analogue of the derivative martingale and prove its convergence to a non-trivial limit. Fixed points multivariate smoothing transform: critical case where d = means same law, then we call the law L (X) of X a fixed point of the multivariate smoothing transform (associated with (T i ) i≥1 ). By a slight abuse of notation, we will also call X a fixed point. This notion goes back to Durrett and Liggett [20] . For d = 1, it is known that properties of fixed points are encoded in the function m(s) are nonnegative random numbers): If some non-lattice and moment assumptions are satisfied, then the existence of an α ∈ (0, 1] with m(α) = 1 and m (α) ≤ 0 is equivalent to the existence of fixed points which then are unique up to scaling. See [30, Theorem 1.1] and [4, Theorem 6.1(a)] for more precise statements of necessary conditions for the existence of fixed points. Moreover, if ψ(r) = E e −rX is the Laplace transform of a fixed point, then there is a positive function L, slowly varying at 0, and K > 0 such that The function L is constant if m (α) < 0 and L(t) = (|log t| ∨ 1) if m (α) = 0, the latter being called the critical case. For α < 1, the property (1.2) implies that the fixed points have Pareto-like tails with index α, i.e. lim t→∞ t −α P (X > t) /L(1/t) ∈ (0, ∞), see [30] for details. Tail behavior in the case α = 1, in which there is no such implication, is investigated in [23, 30, 16] . Existence and uniqueness results in the multivariate setting d ≥ 2 for the noncritical case have been recently proved in [32] . The aim of this work is to provide the corresponding result for the multivariate critical case. In order to so, we will first review necessary notation and definitions from [32], in particular introducing the multivariate analogue of the function m, as well as a result about the existence of fixed points in the critical case. Following the approach in [8, 10, 28] we will then prove that a multivariate regular variation property similar to (1.2) holds for fixed points (with an essentially unique, but yet undetermined slowly varying function L), which we use in order to prove the uniqueness of fixed points, up to scalars. Under some extra (density) assumption, we identify the slowly varying function to be the logarithm also in the multivariate case, which allows us to introduce and prove convergence of the multivariate version of the so-called derivative martingale, a notion coined in [9] . It appears prominently in the limiting distribution of the minimal position in branching random walk, see [1, 2, 9, 14] for details and further references. Our results can be interpreted in the setting of multi-type branching random walk as follows: Consider a particle positioned at e 1 := (1, 0, · · · , 0) ∈ R d ≥ , which produces a random number N of offspring, which are placed at positions x i := T i e 1 . This first generation produces offspring independently in the same manner: The j-th particle has N j children, which are being placed at positions x ji := T i (j)x j , where (N j , (T i (j)) i∈N ) are copies of (N, (T i ) i∈N ). Denote by S ≥ := S d−1 ∩ R d ≥ the intersection of the unit sphere with the cone of vectors with nonnegative entries. Now writing the particle positions x v = u v e sv in logarithmic polar coordinates with u v ∈ S ≥ and s v ∈ R, we observe that the logarithmic distances s v of the particles from the origin perform a multitype branching random walk, with the law of the increments s ji − s j = log |T i (j)u j | depending on the spherical position u j of the ancestor, thus the type space is given by S ≥ . Eq. (1.1) was studied for multitype branching random walks with finite type spaces in [11, 29] and in a two-type setup with type space {−1, 1} in [24, Section 2.6]. Note that our setting is not as general as it seems, for the increment laws depend continuously on the ancestors position u v . Nevertheless, to the best of our knowledge, [24] is the only other reference where the functional equation of the multitype branching random walk in the critical case is studied.

doi:10.1214/ejp.v20-4022
fatcat:bc4dhlfs3rau3c7ybtrsqd7yua