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- W1569456642 abstract "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." @default.
- W1569456642 created "2016-06-24" @default.
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- W1569456642 date "2014-09-25" @default.
- W1569456642 modified "2023-09-27" @default.
- W1569456642 title "Fixed Points of the Multivariate Smoothing Transform: The Critical Case" @default.
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