## A Superlacunary Ito-Kawada Theorem, and Applications to the Equidistribution of Generalized Rudin-Shapiro Polynomials

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##### Date

##### Authors

Cloutier, Daniel

##### Keyword

Random Walks , Rudin-Shapiro Polynomials , Compact Groups , Lie Groups , Lie Algebras , Lacunary Sequences , Superlacunary Sequences

##### Abstract

The Ito-Kawada Theorem is a classical result in the theory of random walks on compact groups. It states that essentially the only obstruction to the equidistribution of a random walk with independent and identically distributed increment is if the support of the increments is confined to proper, closed subgroup of G, or a coset of a proper, closed, normal subgroup of G. We prove a version of the Ito-Kawada theorem for a class of weakly dependent random variables arising from superlacunary sequences. We then show that a conjecture of Doche about the even moments of generalized Rudin-Shapiro polynomials follows from a conjectured Ito-Kawada theorem for lacunary random variables.