Limit Theorems for Theta Sums
نویسنده
چکیده
For almost all values of x 2 R, the classical theta sum S N (x) = N ?1=2 N X n=1 e 2i n 2 x exhibits an extremely irregular behaviour, as N tends to innnity. This limit is investigated by exploiting a connection to the ergodic properties of ows on the unit tangent bundle of a surface of constant negative curvature.
منابع مشابه
Central Limit Theorems for Sums of Wick Products of Stationary Sequences
We show, by the method of cumulants, that checking whether the central limit theorem for sums of Wick powers of a stationary sequence holds can be reduced to the study of an associated graph problem (see Corollary 1). We obtain thus central limit theorems under various integrability conditions on the cumulant spectral functions (Theorems 2, 3). 1 A. Introduction. Let X¡ be a zero mean stationar...
متن کاملConditioned Limit Theorems for Weighted Sums of Random Sequence
Some notions of conditionally dominated random variables are introduced and characterized, Under rather minimal assumptions on random variables {X,Xn, n ≥ 1}, some limit theorems of Jamison’s type of weighted sums of random variables are obtained. Mathematics Subject Classification: 60F15
متن کاملA.D.Wentzell (Tulane University) Limit theorems with asymptotic expansions for stochastic processes. There is a vast riches of limit theorems for sums of independent random variables: theorems about weak convergence, on large deviations, theorems with asymptotic expan-
There is a vast riches of limit theorems for sums of independent random variables: theorems about weak convergence, on large deviations, theorems with asymptotic expansions, etc. We can try to obtain the same kinds of theorems for families of stochastic processes. If X1, X2, ..., Xn, ... is a sequence of independent identically distributed random variables with expectation EXi = 0 and variance ...
متن کاملDiscrete Rough Paths and Limit Theorems
Abstract. In this article, we consider limit theorems for some weighted type random sums (or discrete rough integrals). We introduce a general transfer principle from limit theorems for unweighted sums to limit theorems for weighted sums via rough path techniques. As a by-product, we provide a natural explanation of the various new asymptotic behaviors in contrast with the classical unweighted ...
متن کاملA Multi-variable Theta Product
A multi-variable theta product is examined. It is shown that, under very general choices of the parameters, the quotient of two such general theta products is a root of unity. Special cases are explicitly determined. The second main theorem yields an explicit evaluation of a sum of series of cosines, which greatly generalizes one of Ramanujan’s theorems on certain sums of hyperbolic cosines.
متن کامل