Estimating the center of symmetry: Is it always better to use larger sample sizes?

We show that for symmetric stable distributions with index γ<1, the performance of the sample mean, as an estimator of the center of symmetry, worsens as the sample size increases, where performance is measured by concentration around the estimand. We look at order statistics and show that for symmetric distributions (not necessarily stable), the performance of the sample median as an estimator of the center of symmetry improves as the sample size increases.