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首页> 外文期刊>Canadian Journal of Mathematics >The Central Limit Theorem for Subsequences in Probabilistic Number Theory
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The Central Limit Theorem for Subsequences in Probabilistic Number Theory

机译：概率数论中子序列的中心极限定理

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摘要

Let (n_k)_(k≥1) be an increasing sequence of positive integers, and let f(x) be a real function satisfying (1)f(x +.1) =f(x),∫_0~1(x) dx = 0,Var_([0,1])f< ∞. If lim_(k→ ∞) n_(k+1)_k = ∞ the distribution of (2)∑_(k=1)~N f(n_k-x)/√N converges to a Gaussian distribution. In the case 1 < lim inf k→ ∞ n(k+1)_k,lim supk→ ∞ n(k+1)_k<∞ there is a complex interplay between the analytic properties of the function f, the number-theoretic properties of (n_k)_(k≥l),and the limit distribution of (2). In this paper we prove that any sequence (n_k)_(k≥1) satisfying lim sup_(k →∞)n_(k+1)_k = 1 contains a nontrivial subsequence (m_k)_(k≥1) such that for any function satisfying (1) the distribution of converges to a Gaussian distribution. This result is best possible: for any ε > 0 there exists a sequence (n_k)_(k≥l) satisfying lim sup_(k →∞)n_(k+1)_k≤ 1 + ε such that for every nontrivial subsequence (m_k)_(k≥1) of (n_k)_(k≥1) the distribution of (2) does not converge to a Gaussian distribution for some f. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.
机译：令（n_k）_（k≥1）为正整数的递增序列，令f（x）为满足（1）f（x +.1）= f（x），∫_0〜1（ x）dx = 0，Var _（[0,1]）f <∞。如果lim_（k→∞）n_（k + 1）/ n_k =∞，则（2）∑_（k = 1）〜N f（n_k-x）/√N的分布收敛为高斯分布。在1 0，存在满足lim sup_（k→∞）n_（k + 1）_k≤1 +ε的序列（n_k）_（k≥l），从而对于每个非平凡子序列（n_k）_（k≥1）的（m_k）_（k≥1），对于某些f，（2）的分布不收敛为高斯分布。我们的结果可以看成是拉姆齐类型的结果：一个足够密集的递增整数序列包含一个具有某些要求的数论性质的子序列。

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