# Converting a sequence of dependent random variables to an iid sequence

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.

• 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. – mlofton Mar 15 at 13:57
• Can you please transform your latex notation to visual? (e.g. use dollar signs) – Stats Mar 15 at 15:59