Central Limit Theorem for simultaneous sums

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 $$S^n_2$$?

• $\Sigma$ is the covariance matrix of what? Is each $X_i$ real valued or is it a random vector in $\mathbb R^n$? – Iosif Pinelis Mar 7 at 20:58
• 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 $n$? – Iosif Pinelis Mar 7 at 21:46
• @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. – student Mar 8 at 0:30
• 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). – Anthony Quas Mar 9 at 12:50
• @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? – student Mar 9 at 13:02