Lacunary Series and Independent Functions

Abstract

We review the theoretical and experimental results on semi-leptonic and rare B decays presented in working group 2 of the UK phenomenology workshop on heavy flavour physics and CP violation in Durham, 2000.

On the law of the iterated logarithm for the discrepancy of lacunary sequences

“…The probabilistic analogy is, however, not complete: the asymptotic properties of the sequence n k x depend also on the number-theoretic properties of the sequence (n k ) in an essential way. For example, Kac [12] showed that in the case n k = 2 k the sequence f (n k x) satisfies the central limit theorem for all "nice" periodic functions f , and Gaposhkin [10] showed that this remains valid if the fractions n k+1 /n k are integers or if n k+1 /n k → α, where α r is irrational for r = 1, 2, . .…”

“…There is an extensive literature dealing with the asymptotic properties of the sequence n k x (see Gaposhkin [10] for a survey until 1966), but the precise order of magnitude of its discrepancy has been found only for a few special sequences (n k ). R.C.…”

On the Uniform Theory of Lacunary Series

“…Morgenthaler [43] and Weiss [52] showed that the analogue of (1.2) and (1.3) holds for any uniformly bounded orthonormal system except that one needs a much stronger gap condition for (n k ), the Gaussian limit distribution in (1.2) becomes mixed Gaussian and the limsup in (1.3) becomes a nonconstant function of x. As shown by Gaposhkin [32], at the cost of introducing an extra centering factor, even the orthogonality can be dropped here. Specifically, he proved (see also Chatterji [23], [24]) that if a sequence (X n ) of r.v.…”

“…This result closes a long series of investigations and, together with a wide asymptotic theory of "concrete" (e.g. Hadamard lacunary) subsequences of classical function systems such as the trigonometric and Walsh system, orthogonal polynomials, dilated series ∑ c k f (kx) (for a survey see Gaposhkin [32]), gives a quite satisfactory picture of the asymptotic behavior of lacunary series. Despite this fact, some important question in the field remain open.…”

“…A further example is the result of Gaposhkin [32] stating that for every uniformly bounded sequence (X n ) of random variables there exists a subsequence (X n k ) and random variables X and Y ≥ 0 such that if (a k ) is a sequence of real numbers satisfying…”