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Let $(a_n)_{n \geq 1}$ be a nonnegative, strictly decreasing sequence with $a_n \to 0$ as $n \to \infty$.

Consider the ratio (for $k \geq n$) $$ R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k-1})} \Big[\frac{a_n u_n^2}{\sum_{m=1}^k a_m u_m^2} \Big]. $$ Is it possible to compute the limit $$ R^\star_n = \lim_{k \to \infty} R_{n, k}? $$

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  • $\begingroup$ In the limit you consider, the coordinates become iid centered Gaussians. $\endgroup$
    – fedja
    Commented Aug 5 at 22:21
  • $\begingroup$ Yes. The expectation of that will be the limit of your expectation as $k\to\infty$. $\endgroup$
    – fedja
    Commented Aug 5 at 23:20
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    $\begingroup$ Perhaps a way to restate what you are saying is that (say by dominated convergence), $$ R_n^\star = \mathbb{E} \Big[\frac{a_n g_n^2}{\sum_{m=1}^\infty a_m g_m^2}\Big],$$ where the expectation is taken over the sequence $\{g_m\}_{m \geq 1}$ of iid standard Normals. $\endgroup$
    – Drew Brady
    Commented Aug 5 at 23:29

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