I have a sequence $L = X_1, X_2, ..., X_m$ of iid random variables and another sequence $R = X_{m+1}, X_{m+2}, ..., X_{n}$ of iid random variables such that each $X_i$, for $1 \leq i \leq m$, is possibly dependent on the sequence $R$. For simplicity, assume that the r.vs are all binary.

Question: (a) Does shuffling (using a uniformly random permutation) ($L$, $R$) would make the new sequence iid? That is, suppose I define a new sequence $Y_1$, $Y_2$, $\ldots$, $Y_n$ as follows: (1) first choose a uniformly random permutation $\pi: [n] \rightarrow [n]$. Then (2) define $Y_i = X_{\pi(i)}$. Is the sequence $Y_1$, $\ldots$, $Y_n$ iid?

(b) If instead, I define a new sequence $Y_1$, $Y_2$, $\ldots$, $Y_{\ell}$ as follows: for each $1 \leq i \leq \ell$, $Y_i$ is defined by (1) first choosing a uniformly random $j$ from $[n]$ and (2) letting $Y_i = X_j$. Is the sequence $Y_1$, $Y_2$, $\ldots$, $Y_{\ell}$ iid for any $\ell$?

For simplest cases, we can show that the answer to (a) is False. For instance, say $L = X_1$ and $R = X_2$ be one element sequences. Then $Y_1$, $Y_2$ could be $X_1$, $X_2$ or $X_2$, $X_1$. In either case, knowing $Y_2$ could reveal information about $Y_1$. Hence, the sequence is not independent. However, (b) is true for this case.

On the other hand, if I set $\ell = n+1$ in (b), then the sequence $Y_1$, $\ldots$, $Y_{\ell}$ in (b) is not independent by pigeon-hold argument.

Also, it can shown that the sequence $Y_i$'s are identically distributed. So, the question is about the independence of the resulting sequence. In particular, I am looking for: are there choices of $m$, $n$, and $\ell$ such that the sequence $Y_1$, $Y_2$, $\ldots$ in (a) or in (b) is independent.

  • $\begingroup$ it's an interesting question and I don't know the answer but check out transfer functions ( in time domain, although they may be helpful in frequency domain also ? ) in the box-jenkins-ARIMA framework. That may be what you're interested in but I'm not totally sure. Your problem sounds more complex possibly. $\endgroup$ – mlofton Mar 15 at 13:57
  • $\begingroup$ Can you please transform your latex notation to visual? (e.g. use dollar signs) $\endgroup$ – Stats Mar 15 at 15:59

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