论文标题
对具有共同任意余量的成对独立随机变量的中央限制定理的反例
A counterexample to the central limit theorem for pairwise independent random variables having a common arbitrary margin
论文作者
论文摘要
中心限制定理（CLT）是统计中最基本的结果之一。它指出，$ n $相互独立和相同分布的随机变量的标准化样本平均值，有限的第一和第二矩分配为标准高斯，因为$ n $转移到无限。特别是，序列的成对独立性通常不足以使定理保持。我们明确构建具有共同但任意边缘分布$ f $（满足非常温和的条件）的成对独立随机变量，该变量未经验证。我们通过以封闭形式获得我们序列样品平均值的渐近分布来研究CLT的“失败”程度。通过几个理论示例来说明这一点，我们为此提供了R语言相关的计算代码。
The Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of $n$ mutually independent and identically distributed random variables with finite first and second moments converges in distribution to a standard Gaussian as $n$ goes to infinity. In particular, pairwise independence of the sequence is generally not sufficient for the theorem to hold. We construct explicitly a sequence of pairwise independent random variables having a common but arbitrary marginal distribution $F$ (satisfying very mild conditions) for which the CLT is not verified. We study the extent of this 'failure' of the CLT by obtaining, in closed form, the asymptotic distribution of the sample mean of our sequence. This is illustrated through several theoretical examples, for which we provide associated computing codes in the R language.
