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Context: While trying to teach the Central Limit Theorem I thought it would be a good idea to show a case where it breaks down.

Question: Consider the sum of increasing powers of standard normal random variables. So $Y= X_1 + X_2^2 + X_3^3 + X_4^4 + \dotsm + X_n^n$ where $X_i \sim N(0,1)$. If we let $n\rightarrow \infty$ does $Y$ converge to any known distribution?

My preliminary simulations make me think it looks like a Cauchy distribution but I have not been able to find any literature on what $Y$ might be.

kjetil b halvorsen
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S. Punky
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  • Thanks. Are they also supposed to be independent? – kjetil b halvorsen Oct 31 '18 at 18:01
  • Assuming you are teaching the classical version of CLT, we need to assume that the variables being summed are iid with finite mean and variance. You don't have identical distributions here. Instead, you could use $X_i \stackrel{iid}{\sim} t(v)$ for $v \leq 2$. Or for a more interesting example, consider the St Petersburg paradox where $X_i \sim Geometric(1/2)$ and $Y_i = 2^{X_i}$. – knrumsey Oct 31 '18 at 18:02
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    Clearly $Y_n$ does not converge at all--but shall we presume you wish, as usual, to standardize $Y_n$? If so, notice how the behaviors of $Y_{n}$ and $Y_{n+1}$ differ so markedly and consider the super-exponential growth rate of their variances. That shows even the standardized $Y_n$ cannot converge. I believe you can obtain convergence to (very simple) distributions if you confine the sum to $X_i^i$ with (eventually) all odd or all even values of $i.$ – whuber Oct 31 '18 at 18:03
  • A simple case where CLT breaks down is at https://stats.stackexchange.com/questions/192652/central-limit-theorem-and-the-pareto-distribution/192717#192717 – kjetil b halvorsen Dec 13 '23 at 21:42

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