两两独立随机变量具有公共任意边界的中心极限定理的一个反例
A counterexample to the central limit theorem for pairwise independent random variables having a common arbitrary margin
摘要：
中心极限定理是统计学中最基本的结果之一。它指出，一个序列的$n$相互独立和同分布的随机变量与有限的第一和第二时刻的标准化样本均值收敛到一个标准高斯分布的$n$走向无穷大。特别是，序列的两两独立性通常不足以使定理成立。我们明确地构建了一个序列的成对独立的随机变量具有一个共同的，但任意的边缘分布$F$（满足非常温和的条件）的CLT是没有验证。我们研究的程度，这种“失败”的CLT获得，在封闭的形式，渐近分布的样本均值，我们的序列。这是通过几个理论的例子，我们提供了相关的计算代码在R语言说明。
英文摘要：
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.
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