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### How to calculate this limit (if exist)?

$\newcommand\ka\kappa$The problem obviously reduces to this one: find $$L:=\lim_{m\to\infty}\frac{S(c,m-1,n_1-1,n_2-1)}{S(c,m,n_1,n_2)},$$ where $c:=a/b\ne1$ and \begin{...
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• 82.4k
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### Behavior of F distribution quantile as degree of freedom varies

$\newcommand\al\alpha$If the $(1-\al)$-quantiles of $F_{n,m}$ were decreasing monotonically in $m$ for each $\al\in(0,1)$, the the corresponding cdf's -- say $G_{m,n}$ -- would be increasing pointwise ...
• 82.4k
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### Triangle equality for cosine similarity in high dimensions

$\newcommand{\R}{\mathbb R}$Here is a straightforward explanation in the case of adding iid standard normal random variables (r.v.'s). Here we have random vectors U:=X,\quad V:=X+...
• 82.4k
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• 82.4k

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