Fixed Points of the Multivariate Smoothing Transform: The Critical Case

Given a sequence $(T_1, T_2, ...)$ of random $d \times d$ matrices with nonnegative entries, suppose there is a random vector $X$ with nonnegative entries, such that $ \sum_{i \ge 1} T_i X_i $ has the same law as $X$, where $(X_1, X_2, ...)$ are i.i.d. copies of $X$, independent of $(T_1, T_2, ...)$. 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 $m$ is introduced, such that the existence of $\alpha \in (0,1]$ with $m(\alpha)=1$ and $m'(\alpha) \le 0$ guarantees the existence of nontrivial fixed points. We prove the uniqueness of fixed points in the critical case $m'(\alpha)=0$ 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.