Almost Sure Bootstrap of the Mean Under Random Normalization

Abstract

We consider the problem of when the bootstrap sample mean, appropriately normalized and centered, converges in distribution along almost every sample path. We allow the normalizing sequence to be an arbitrary sequence of positive random variables. It is proved that the only possible normalizing sequence is essentially $(\sum^n_{i = 1}X^2_i)^{1/2}$. Furthermore, if the bootstrap sample mean converges along almost every sample path, then either the variance is finite or else the distribution of $X$ is extremely heavy tailed. In the latter case, the distribution of the bootstrap sample mean is completely determined by how many times the maximum order statistic from the original random sample is repeated in the bootstrap sample. The necessary condition on how heavy the tails must be is $(\sum^n_{i = 1}|X_i|^p)^{1/p}/(\sum^n_{i = 1}X^2_i)^{1/2} \rightarrow 1$ almost surely for all $p \in (0, \infty\rbrack$. Furthermore, we show that in this case the limit of the bootstrap sample mean normalized by $(\sum^n_{i = 1}X^2_i)^{1/2}$ is Poisson with mean 1.