Dual Lukacs regressions for non-commutative variables

Kamil Szpojankowski , Jacek Wesołowski


Dual Lukacs type characterizations of random variables in free probability are studied here. First, we develop a freeness property satisfied by Lukacs type transformations of free-Poisson and free-Binomial non-commutative variables which are free. Second, we give a characterization of non-commutative free-Poisson and free-Binomial variables by properties of first two conditional moments, which mimic Lukacs type assumptions known from classical probability. More precisely, our result is a non-commutative version of the following result known in classical probability: if \$U\$, \$V\$ are independent real random variables, such that \$E(V(1-U)\\textbar\UV)\$ and \$E(V\\textasciicircum\2(1-U)\\textasciicircum\2\\textbar\UV)\$ are non-random then \$V\$ has a gamma distribution and \$U\$ has a beta distribution.
Author Kamil Szpojankowski
Kamil Szpojankowski,,
, Jacek Wesołowski (FMIS / DPMS)
Jacek Wesołowski,,
- Department of Probability and Mathematical Statistics
Journal seriesarXiv:1110.3419, (0 pkt)
Issue year2011
Keywords in English46L54 (Primary), 62E10 (Secondary), Mathematics - Operator Algebras, Mathematics - Probability
URL http://arxiv.org/abs/1110.3419
Score (nominal)0
Citation count*1 (2013-01-30)
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