论文标题
cauchy，正常和相关性与重型尾巴
Cauchy, normal and correlations versus heavy tails
论文作者
论文摘要
Pillai and Meng（2016）的令人惊讶的结果表明，转换$ \ sum_ {j = 1}^n w_j x_j/y_j $的两个IID中心的正常随机向量，$（x_1，\ ldots，x_n）$和$（y_1，y_1，y__n）$，$ n> $ $， $ j = 1，\ ldots，n $，$ \ sum_ {j = 1}^n w_j = 1 $，无论普通向量内的任何相关性如何。在与重型尾巴的比赛中，相关性似乎失去了。为了阐明这种现象的广泛程度，我们分析了两个IID中心正常随机向量的另外两个转换。这些转变在精神上与Pillai和Meng（2016）所考虑的转变相似。一个转换涉及绝对值：$ \ sum_ {j = 1}^n w_j x_j/| y_j | $。第二个涉及随机停止布朗动议：$ \ sum_ {j = 1}^n w_j x_j \ bigl（y_jj^{ - 2} \ bigr）$，其中$ \ bigl \ {\ bigl \ {\ bigl（x_1（x_1（x_1（x_1（t）），\ ldots，\ ldots，\ ldots，x_n（t） n> 1，$是一个具有正差异的布朗运动； $（y_1，\ ldots，y_n）$是一个中心的正常随机向量，其定律与$（x_1（1），\ ldots，x_n（1））$相同，并且独立于此； $ x（y^{ - 2}）$是在随机时间评估的布朗运动$ x（t）$的值$ t = y^{ - 2} $。如果正常成分的协方差矩阵是对角线的，或者如果协方差矩阵等于1，则所有三个转换都会导致cauchy分布。但是，如果转化Pillai and Meng（2016）认为产生了cauchy分布，而与正常的共证量矩阵无关。我们在这里考虑的转换并不总是会产生库奇分布。边缘尾巴的重度并不总是淹没共同正常随机变量之间的相关性。正常法律和库奇法律之间的联系的奥秘仍有待理解。
A surprising result of Pillai and Meng (2016) showed that a transformation $\sum_{j=1}^n w_j X_j/Y_j$ of two iid centered normal random vectors, $(X_1,\ldots, X_n)$ and $(Y_1,\ldots, Y_n)$, $n>1$, for any weights $0\leq w_j\leq 1$, $ j=1,\ldots, n$, $\sum_{j=1}^n w_j=1$, has a Cauchy distribution regardless of any correlations within the normal vectors. The correlations appear to lose out in the competition with the heavy tails. To clarify how extensive this phenomenon is, we analyze two other transformations of two iid centered normal random vectors. These transformations are similar in spirit to the transformation considered by Pillai and Meng (2016). One transformation involves absolute values: $\sum_{j=1}^n w_j X_j/|Y_j|$. The second involves randomly stopped Brownian motions: $\sum_{j=1}^n w_j X_j\bigl(Y_j^{-2}\bigr)$, where $\bigl\{\bigl( X_1(t),\ldots, X_n(t)\bigr), \, t\geq 0\bigr\},\ n>1,$ is a Brownian motion with positive variances; $(Y_1,\ldots, Y_n)$ is a centered normal random vector with the same law as $( X_1(1),\ldots, X_n(1))$ and independent of it; and $X(Y^{-2})$ is the value of the Brownian motion $X(t)$ evaluated at the random time $t=Y^{-2}$. All three transformations result in a Cauchy distribution if the covariance matrix of the normal components is diagonal, or if all the correlations implied by the covariance matrix equal 1. However, while the transformation Pillai and Meng (2016) considered produces a Cauchy distribution regardless of the normal covariance matrix. the transformations we consider here do not always produce a Cauchy distribution. The correlations between jointly normal random variables are not always overwhelmed by the heaviness of the marginal tails. The mysteries of the connections between normal and Cauchy laws remain to be understood.
