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A law of the iterated logarithm for general lacunary series

Charles N. Moore, Xiaojing Zhang (2012)

Colloquium Mathematicae

We prove a law of the iterated logarithm for sums of the form k = 1 N a k f ( n k x ) where the n k satisfy a Hadamard gap condition. Here we assume that f is a Dini continuous function on ℝⁿ which has the property that for every cube Q of sidelength 1 with corners in the lattice ℤⁿ, f vanishes on ∂Q and has mean value zero on Q.

A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin (2013)

Colloquium Mathematicae

We prove the central limit theorem for the multisequence 1 n N 1 n d N d a n , . . . , n d c o s ( 2 π m , A n . . . A d n d x ) where m s , a n , . . . , n d are reals, A , . . . , A d are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in [ 0 , 1 ] s . The main tool is the S-unit theorem.

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