I read through an example in which the author states that the following application of law of large numbers (without proof or explanation):

"Let $X_1,X_2,...$ be i.i.d. random variables such that $E|X_1|^k<\infty$.

Then $\frac{X_1^k+X_2^k+...+X_n^k}{n} \overset{p}{\to}E(X_1^k)$ as $n \to \infty$."

Could someone kind enough explain this to me?


This follows from the law of large numbers applied to the variables $X_i^k$ in place of the usual $X_i$.

The reason this is the case is that if the $X_i$ form an i.i.d. sequence, then also the $X_i^k$ will form an i.i.d. sequence, and moreover if $\mathbb E|X_1|^k<\infty$, then the variables $X_i^k$ will have finite absolute mean, which suffices for the law of large numbers to apply. Note that the same argument works for both modes of convergence (weak law and strong law).

  • $\begingroup$ In fact, convergence holds in almost sure sense. $\endgroup$ – Kavi Rama Murthy Sep 28 '18 at 8:30
  • $\begingroup$ "as required" should be "as is sufficient" $\endgroup$ – mathworker21 Sep 28 '18 at 8:35
  • $\begingroup$ Thanks for the nitpicks, I updated my answer to incorporate both :) $\endgroup$ – pre-kidney Sep 28 '18 at 8:39

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