## New answers tagged st.statistics

3
votes

Accepted

### 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{...

- 82.4k

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
$$
\...

- 10.2k

0
votes

Accepted

### 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 $...

- 21.7k

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 $...

- 82.4k

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 ...

- 82.4k

4
votes

Accepted

### $\min(\det(\mathbf{A}))$ 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 ...

- 72.6k

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 ...

- 72.6k

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 $X-\mu$...

- 82.4k

3
votes

Accepted

### 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

2
votes

Accepted

### 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+...

- 82.4k

2
votes

Accepted

### 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 ...

- 82.4k

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