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 $S^n_2$?

  • $\begingroup$ $\Sigma$ is the covariance matrix of what? Is each $X_i$ real valued or is it a random vector in $\mathbb R^n$? $\endgroup$ – Iosif Pinelis Mar 7 at 20:58
  • $\begingroup$ What do you mean by "For the vector $a \in \mathbb{R}^n$ of unit length $\Vert a \Vert = 1$, assume the CLT $$a^T \Sigma^{-1/2} X \rightsquigarrow N(0,1)$$ holds for sufficiently large sample sizes."? How do the distributions of the $X_i$'s depend on $n$? How does the vector $a$ depend on $n$? How do the vectors $a_i$ depend on $n$? $\endgroup$ – Iosif Pinelis Mar 7 at 21:46
  • $\begingroup$ @IosifPinelis I have tried to come up with more a precise definition of what I want to achieve. The idea is to take many different sums of $\pm X_i$ simultaneously in a way that, if they were jointly normal, they would also be independent. But on a more basic level, I want to know if two sums of $\pm X_i$ tend to a jointly normal distribution. $\endgroup$ – student Mar 8 at 0:30
  • $\begingroup$ I don’t think the question makes sense as written (even if you restrict to the Hadamard case). The problem is: what do you mean by the limit of $H_nX/s_n$? These are random variables taking values in spaces of different dimensions (so it makes no sense to take a limit of them). $\endgroup$ – Anthony Quas Mar 9 at 12:50
  • $\begingroup$ @AnthonyQuas True, that is because I was thinking of a large but finite sample. To take a limit, one could restrict to the first $k$ columns of $H_n$: does $H_n^{(k)} X/s_n$ tend to $N(0, I_k)$? But take any convenient prescription for the $\pm X_i$ terms: what is the joint distribution of different sums? $\endgroup$ – student Mar 9 at 13:02

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