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Let $\{X_n\}_{n=1}^\infty$ be a sequence of iid random variables under the Cauchy distribution with location parameter $0$ and scaling parameter $1$ (so that the density function is $f(x) = \frac{1}{\pi(1+x^2)}$).

It can be shown that the characteristic function of these random variables will look like $\varphi(t) = e^{-|t|}$, so $\varphi$ is not differentiable at 0, and hence the weak law of large numbers cannot be applied (because the first moment is infinite).

However, how would one go about actually showing that the result of the weak law isn't true (assuming it isn't)? In other words, how can one show that there does not exist a constant $b$ such that $\frac{1}{N} \sum_{n=1}^N X_n \stackrel{\mathbb{P}}{\to} b$? (Here $\stackrel{\mathbb{P}}{\to}$ denotes convergence in probability).

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The finiteness of the first moment is not a necessary condition for the weak law of large numbers. – Cm7F7Bb Sep 30 '14 at 8:46
up vote 4 down vote accepted

$\overline{X}_N = \frac{1}{N} \sum_{n=1}^N X_n$ also has the Cauchy distribution with parameters $0$ and $1$: this can easily be seen from the characteristic function. In particular it doesn't approach a constant in probability.

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