Abstract
By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d. sequences. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis, such as the theory of orthogonal series and Banach space theory. In contrast to i.i.d. sequences, however, the probabilistic structure of lacunary sequences is not permutation-invariant and the analytic properties of such sequences can change after rearrangement. In a previous paper we showed that permutation-invariance of subsequences of the trigonometric system and related function systems is connected with Diophantine properties of the index sequence. In this paper we will study permutation-invariance of subsequences of general r.v. sequences.
Highlights
It is known that sufficiently thin subsequences of general r.v. sequences behave like i.i.d. sequences
In this paper we will prove the surprising fact that, in a sense to be made precise, all nonparametric distributional limit theorems for i.i.d. random variables hold for lacunary subsequences of general r.v. sequences in a permutation-invariant form provided that the subsequence is sufficiently thin, i.e. the gaps of the sequence grow sufficiently rapidly. We will deduce this result from a general structure theorem for lacunary sequences proved in [6] stating that sufficiently thin subsequences of any tight sequence of random variables are nearly exchangeable
Using the terminology of [7], we call a sequence (Xn) of random variables determining if it has a limit distribution relative to any set A in the probability space with
Summary
It is known that sufficiently thin subsequences of general r.v. sequences behave like i.i.d. sequences. We will deduce this result from a general structure theorem for lacunary sequences proved in [6] stating that sufficiently thin subsequences of any tight sequence of random variables are nearly exchangeable. Using the terminology of [7], we call a sequence (Xn) of random variables determining if it has a limit distribution relative to any set A in the probability space with
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