Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory
Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory
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因随机变量的经验分布函数和强近似定理。
DOI：
10.1090/s0002-9947-1994-1249469-5
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发表时间：
1994
影响因子：
1.3
通讯作者：
W. Philipp
中科院分区：
文献类型：
--
作者：
W. Philipp
． 设y = {ii，…，4t}是一个有限的素数整数集，设{»1，«2，…}表示由y生成的乘法半群，并按递增顺序排列。令Dn(w)表示序列{nkco}%=i mod 1, w e[0,1]的差值。本文通过证明除了Lebesgue测度0 l以外的所有w <limsuP /™*(ft,) <C，解决了R.C. Baker[3]提出的一个问题。这里常数C只取决于q的质因数分解所涉及的素数的总数，…, qx、。由序列{cos2nnk<û}kx>=l的独立标准正态随机变量和的部分和的一个强逼近定理得到了下界。
. Let y = {ii, ... ,4t} be a finite set of coprime integers and let {»l, «2,...} denote the mutiplicative semigroup generated by y, and arranged in increasing order. Let Dn(w) denote the discrepancy of the sequence {nkco}%=i mod 1, w e [0, 1). In this paper we solve a problem posed by R.C. Baker [3], by proving that for all w except on a set of Lebesgue measure 0 l<limsuP /™*(ft,) <C. Here the constant C only depends on the total number of primes involved in the prime factorization of q¡ ,... , qx. The lower bound is obtained from a strong approximation theorem for the partial sums of the sequence {cos2nnk<û}kx>=l by sums of independent standard normal random variables.