653 research outputs found
Limit law for some modified ergodic sums
An example due to Erdos and Fortet shows that, for a lacunary sequence of
integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of
normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a
generalization of their example interpreted as the limiting behavior of some
modified ergodic sums in the framework of dynamical systems
Factoring bivariate lacunary polynomials without heights
We present an algorithm which computes the multilinear factors of bivariate
lacunary polynomials. It is based on a new Gap Theorem which allows to test
whether a polynomial of the form P(X,X+1) is identically zero in time
polynomial in the number of terms of P(X,Y). The algorithm we obtain is more
elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on
the valuation of polynomials of the previous form instead of the height of the
coefficients. As a result, it can be used to find some linear factors of
bivariate lacunary polynomials over a field of large finite characteristic in
probabilistic polynomial time.Comment: 25 pages, 1 appendi
On permutations of lacunary series
It is a well known fact that for periodic measurable and rapidly
increasing the sequence behaves like a
sequence of independent, identically distributed random variables. For example,
if is a periodic Lipschitz function, then satisfies
the central limit theorem, the law of the iterated logarithm and several
further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence
remains i.i.d.\ after any permutation of its terms, it is natural to expect
that the asymptotic properties of lacunary series are also
permutation-invariant. Recently, however, Fukuyama (2009) showed that a
rearrangement of the sequence can change substantially its
asymptotic behavior, a very surprising result. The purpose of the present paper
is to investigate this interesting phenomenon in detail and to give necessary
and sufficient criteria for the permutation-invariance of the CLT and LIL for
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