# Convergence of bootstrap standard error estimate (one of the problems from Efron's book)

Cross Validated Asked by Mattjosh on October 26, 2020

I am trying to solve problem 2.5 from the Efron/Tibshirani book, An Introduction to the Bootstrap (page 16).

The problem asks to show that by applying the weak law of large numbers, the bootstrap estimate of standard error

$$hat{se}_{boot} = left{ sum_{b=1}^{B} [s(x^{*b}) – s(cdot)]^2 / (B-1) right}^{1/2}$$

approaches

$$left{ sum_{i=1}^{n} (x_i – bar{x})^2 / n^2 right}^{1/2}$$

when the number of bootstrap samples, $$B$$, goes to infinity.

$$x_i$$ are the elements of the original sample which is of size $$n$$, $$x^{*b}$$ is the b-th bootstrap sample, $$s(cdot) = sum_{b=1}^B s(x^{*b})/B$$, and $$s(x)$$ is the mean $$bar{x}$$.

I can see empirically that this seems to be true, though I wonder how the proof is performed analytically.

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