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3 votes
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How to calculate this limit (if exist)?

$\newcommand\ka\kappa$The problem obviously reduces to this one: find \begin{equation} L:=\lim_{m\to\infty}\frac{S(c,m-1,n_1-1,n_2-1)}{S(c,m,n_1,n_2)}, \end{equation} where $c:=a/b\ne1$ and \begin{...
2 votes

Does the average of correlated Gaussian random variables with mean zero and different variances converge in probability to their mean?

Let $\sigma_{i,j} = \operatorname{cov}(X_i,X_j) $ for $i,j\in\{\,1,2,3,\ldots\,\}.$ (In particular $\sigma_{i,i}= \operatorname{var}(X_i).$ Let $\overline X_n= \dfrac{X_1+\cdots+X_n} n.$ Then $$ \...
0 votes
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WLLN for bootstrap means of stationary ergodic processes?

Answered in comments above It seems as though the answer should be yes. I would suggest writing $X_n$ as $Y_n+Z_n$ where $Y_n$ is $X_n$ if $|X_n|\le m(n)^{1/3}$ and 0 otherwise; similarly $Z_n$ is $...
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2 votes

Does $E[1/f]\overset{d}\to 1/E[f]$ for $\operatorname{Tr}H=1,\operatorname{Tr}H^2=0.5$?

$\newcommand{\tr}{\operatorname{tr}}\newcommand{\de}{\delta} $The posted conditions are not nearly enough for the desired concentration property. For instance, let $Ex=0$ with $H=Exx^\top$ being the $...
4 votes
Accepted

How fast does this Gaussian random walk move away from the origin?

$\newcommand\1{\mathbf1}\newcommand\R{\mathbb R}\newcommand\Si{\Sigma}\newcommand{\si}{\sigma}\newcommand{\Ga}{\Gamma}$Letting $$y_i:=g(z_i)z_i=z_iz_i^\top\1,$$ where $\1:=[1,\dots,1]^\top\in\R^d$, we ...
4 votes
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$\min(\det(\mathbf{A}))$ for special matrix $\mathbf{A}$

Q1 The determinant is $\prod_{n=1}^{N-1} (1 - e^{-2\alpha(p_{n+1}-p_n)})$. Q2 Yes, using the answer to Q1. Q3 Yes, using the answer to Q1. The formula for Q1 is proved by induction on $N$. The ...
4 votes

$\min(\det(\mathbf{A}))$ for special matrix $\mathbf{A}$

To answer Question Q2: Yes, as the special case $u_j = \alpha p_j$ of the following result. We use $j,k$ for the indices rather than $i,j$ because we need $i = \sqrt{-1}$. Proposition. For pairwise ...
3 votes
Accepted

Probabilistic Taylor theorem for concave functions

If $g^{(4)}\le0$, then $$g(x)=\sum_{k=0}^3\frac{g^{(k)}(0)}{k!}\,x^k+\frac{x^4}4\, \int_0^1g^{(4)}(sx)(1-s)^3\,ds \le\sum_{k=0}^3\frac{g^{(k)}(0)}{k!}\,x^k$$ for real $x$. Replacing here $x$ by $X-\mu$...
3 votes
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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 ...
2 votes
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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 \begin{equation} U:=X,\quad V:=X+...
2 votes
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What is a tensor product of random variables?

In this context, I believe the tensor product on random variables is nothing other than the tensor product over the values of the RVs. (In other words, if $\Omega$ is a sample space and $X : \Omega \...
4 votes
Accepted

CLT convergence rate for sum of uniforms (in TV distance)

$\newcommand\dee{\Delta_{\mathrm{TV}}}\newcommand{\vpi}{\varphi}$Let $f_n$ denote the pdf of \begin{equation*} S_n:=\sqrt{\frac3n} \sum_{i=1}^n X_i \end{equation*} and let $\vpi$ denote the ...

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