We derive limiting distributions of symmetrized estimators of scatter, where instead of all $n(n-1)/2$ pairs of the $n$ observations we only consider $nd$ suitably chosen pairs, $1 \le d < \lfloor n/2\rfloor$. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever $d = d(n) \to \infty$ at arbitrarily slow speed. We also investigate the asymptotic properties for arbitrary fixed $d$. These considerations and numerical examples indicate that for practical purposes, moderate fixed values of $d$ between,say, $10$ and $20$ yield already estimators which are computationally feasible and rather close to the original ones.

翻译：我们推导了对称化散布估计量的极限分布，其中使用$n$个观测值中所有$n(n-1)/2$对数据中的$nd$个适当选取的对，$1 \le d < \lfloor n/2\rfloor$。结果表明，当$d = d(n) \to \infty$（即使收敛速度任意慢）时，所得估计量与原估计量渐近等价。我们还研究了任意固定$d$下的渐近性质。这些分析与数值示例表明，在实际应用中，当$d$取适中固定值（例如介于10到20之间）时，所得估计量既具有计算可行性，又与原估计量相当接近。