Real analitic probabilistic approach to theory of random numbers

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1. We proved almost sure invariance principles for lacunary trigonometric series with gap n_<k+1>/n_k【greater than or equal】1+c/k^α(α<1) assuming some condition relating to the convergence of fourth moment. As the collorary of this result, we can derive almost sure invariance principles and hence the classical law of the iterated logarithm for sum Σcos2πn_kt where n_k=[exp(k^β)] (β>4/9), for which P. Erdos made a conjecture.2. We proved the polynomial central limit theorem for non-conventional aveage of Riesz-Raikov sums. The typical examples is Σf(θ^nx)g(θ^n^2x). Sum of this type had been ergodic point of view, but our result gives the probabilistic limit theorems.3. We proved the central limit theorem for Σf(n_kt) where n_k is the Baker's sequence. The Baker's sequence is the increasing arrangement of the set consisting of arbitrary product of some integers belonging the finite set of coprime integers.4. We proved the a.e. version of the law of the large numbers for sum Σf(n_kt) where n_k satisfies the condition of Koksma type.