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Take a sample $X_1 \ldots X_n$ of $n$ independent observations $X_j \in \mathbb{R}$ with zero mean and finite variances $\sigma_j^2$. For $i = 1, 2, \ldots$, define the sums $$S^n_i = \frac{\pm X_1 \pm X_2 \ldots \pm X_n}{s_n} \;,$$ where different $i$'s have different sets of $\pm$ and $s_n^2 = \sum_{j=1}^{n} \sigma_j^2$ is the sum of variances. Assuming suitable conditions on the $X_i$, we have the CLT $$S^n_i \rightsquigarrow N(0,1)$$ for all individual $i = 1 \ldots n$.

Q: What is known about the asymptotic joint distribution of $S^n_i$? Are they jointly normal?

In particular, let $n$ be a number for which the Hadamard matrix $H_n$ exists, and $X = (X_1 \ldots X_n)$. If the distribution were jointly normal, $\frac{H_n X}{s_n}$ would tend towards the multivariate normal $N(0, I_n)$. That's why I suspect that the distribution is either not jointly normal, or the rate of convergence breaks down.

Take a sample $X_1 \ldots X_n$ of $n$ independent observations $X_j \in \mathbb{R}$ with zero mean and finite variances $\sigma_j^2$. For $i = 1, 2, \ldots$, define the sums $$S^n_i = \frac{\pm X_1 \pm X_2 \ldots \pm X_n}{s_n} \;,$$ where different $i$'s have different sets of $\pm$ and $s_n^2 = \sum_{j=1}^{n} \sigma_j^2$ is the sum of variances. Assuming suitable conditions on the $X_i$, we have the CLT $$S^n_i \rightsquigarrow N(0,1)$$ for all individual $i = 1 \ldots n$.

Q: What is known about the asymptotic joint distribution of $S^n_i$? Are they jointly normal?

In particular, let $n$ be a number for which the Hadamard matrix $H_n$ exists, and $X = (X_1 \ldots X_n)$. If the distribution were jointly normal, $\frac{H_n X}{s_n}$ would tend towards the multivariate normal $N(0, I_n)$. That's why I suspect that the distribution is either not jointly normal, or the rate of convergence breaks down.

Edit: We can reduce the question to the most basic case. Let $$S^n_1 = \frac{1}{s_n}\sum_{j=1}^{n} X_j \;, \\ S^n_2 = \frac{1}{s_n} \sum_{j=1}^{n} (-1)^j X_j \;,$$ what is the joint limiting distribution of $S^n_1$ and $S^n_2$?

Take a sample $X_1 \ldots X_n$ of $n$ independent observations $X_j \in \mathbb{R}$ with zero mean and finite variances $\sigma_j^2$. For $i = 1, 2, \ldots$, define the sums $$S^n_i = \frac{\pm X_1 \pm X_2 \ldots \pm X_n}{s_n} \;,$$ where different $i$'s have different sets of $\pm$ and $s_n^2 = \sum_{j=1}^{n} \sigma_j^2$ is the sum of variances. Assuming suitable conditions on the $X_i$, we have the CLT $$S^n_i \rightsquigarrow N(0,1)$$ for all individual $i = 1 \ldots n$.

Q: What is known about the asymptotic joint distribution of $S^n_i$? Are they jointly normal?

In particular, let $n$ be a number for which the Hadamard matrix $H_n$ exists, and let $A = \frac{1}{\sqrt{n}} H_n$ and $\Sigma = \operatorname{diag}(\sigma_1^2 \ldots \sigma_n^2)$. If the distribution were jointly normal, then $A \Sigma^{-1/2} X$ would tend towards the multivariate normal $N(0, I_n)$. That's why I suspect that the distribution is either not jointly normal, or the rate of convergence breaks down.

Take a sample $X_1 \ldots X_n$ of $n$ independent observations $X_j \in \mathbb{R}$ with zero mean and finite variances $\sigma_j^2$. For $i = 1, 2, \ldots$, define the sums $$S^n_i = \frac{\pm X_1 \pm X_2 \ldots \pm X_n}{s_n} \;,$$ where different $i$'s have different sets of $\pm$ and $s_n^2 = \sum_{j=1}^{n} \sigma_j^2$ is the sum of variances. Assuming suitable conditions on the $X_i$, we have the CLT $$S^n_i \rightsquigarrow N(0,1)$$ for all individual $i = 1 \ldots n$.

Q: What is known about the asymptotic joint distribution of $S^n_i$? Are they jointly normal?

In particular, let $n$ be a number for which the Hadamard matrix $H_n$ exists, and $X = (X_1 \ldots X_n)$. If the distribution were jointly normal, $\frac{H_n X}{s_n}$ would tend towards the multivariate normal $N(0, I_n)$. That's why I suspect that the distribution is either not jointly normal, or the rate of convergence breaks down.

Take a sample $X_1 \ldots X_n$ of $n$ independent observations $X_j \in \mathbb{R}$ with zero mean and finite variances $\sigma_j^2$. Let $A$ be a matrix with entries $A = (a_{ij})$ where all $a_{ij} = \pm \frac{1}{\sqrt{n}}$ and no rows are repeated.

For all $i = 1 \ldots n$, the scaled random variable $\frac{a_{ij} X_j}{\sigma_j}$ has variance $a_{ij}^2$, and $\sum_{j=1}^{n} a_{ij}^2 = 1$. Let $$S^n_i = \sum_{j=1}^{n} \frac{a_{ij} X_j}{\sigma_j} \;.$$ Assuming suitable conditions on the $X_i$, we have the CLT $$S^n_i \rightsquigarrow N(0,1)$$ for all individual $i = 1 \ldots n$.

What is known about the asymptotic joint distribution of $S^n_i$? Are they jointly normal?

In particular, let $n$ be a number for which the Hadamard matrix $H_n$ exists, and let $A = \frac{1}{\sqrt{n}} H_n$ and $\Sigma = \operatorname{diag}(\sigma_1^2 \ldots \sigma_n^2)$. If the distribution were jointly normal, then $A \Sigma^{-1/2} X$ would tend towards the multivariate normal $N(0, I_n)$. That's why I suspect that the distribution is either not jointly normal, or the rate of convergence breaks down.

Take a sample $X_1 \ldots X_n$ of $n$ independent observations $X_j \in \mathbb{R}$ with zero mean and finite variances $\sigma_j^2$. For $i = 1, 2, \ldots$, define the sums $$S^n_i = \frac{\pm X_1 \pm X_2 \ldots \pm X_n}{s_n} \;,$$ where different $i$'s have different sets of $\pm$ and $s_n^2 = \sum_{j=1}^{n} \sigma_j^2$ is the sum of variances. Assuming suitable conditions on the $X_i$, we have the CLT $$S^n_i \rightsquigarrow N(0,1)$$ for all individual $i = 1 \ldots n$.

Q: What is known about the asymptotic joint distribution of $S^n_i$? Are they jointly normal?

In particular, let $n$ be a number for which the Hadamard matrix $H_n$ exists, and let $A = \frac{1}{\sqrt{n}} H_n$ and $\Sigma = \operatorname{diag}(\sigma_1^2 \ldots \sigma_n^2)$. If the distribution were jointly normal, then $A \Sigma^{-1/2} X$ would tend towards the multivariate normal $N(0, I_n)$. That's why I suspect that the distribution is either not jointly normal, or the rate of convergence breaks down.