A fixed point theorem for distributions

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A Distributional Perspective on Reinforcement Learning (2017, background): “In general, T π is not a contraction in the p th centered moment, p > 2 , but the centered moments of the iterates { Z k } still converge exponentially quickly to those of Z π ; the proof extends the result of R¨osler (1992).”
Self-similar fragmentations (2002, background): “where = ( j,j 2 N) has the law (·)/ (S # ), and (j) 1 are independent copies of 1, also independent of . It is known that under fairly general conditions, such identity characterizes the law 1 uniquely, see e.g. [ 25 ].”
Random walk on sparse random digraphs (2015, unknown): “A more general version of (7) has been studied by R¨osler [38], Liu [30, 29, 31, 32] and Barral [4, 5].”
A survey of max-type recursive distributional equations (2004, background): “There is a unique solution with E[X2] < ∞ because T is a contraction under the metric d2 at (7) [59].”