Is independence preserved if a random entry in an independent sequence is replaced by a constant?

Question

Let $\xi=(\xi_1,\ldots,\xi_n)$ be a sequence of independent random variables. Let us pick an index $\nu\in \{1,\ldots,n\}$, and replace the entry $\xi_\nu$ by a constant $c$. The rest of the $\xi_i$ remain unchanged.

Question: Is it true that the altered sequence remains independent, even if $\nu$ is random, possibly dependent on $\xi$?

Note: If $\nu$ is a fixed constant index, then the independence is clearly preserved. It is less clear, however, what happens if $\nu$ is random and dependent on $\xi$. For example, we may pick the first hit to some set, or apply any other data-dependent rule to pick the index $\nu$.

Answer 1

The answer is no. E.g., assume that $n=2$, $\xi_i=X_i$, and $X_1,X_2,\nu$ are any random variables (r.v's) each with values in the set $\{1,2\}$ and with $P(\nu=1)=p\in(0,1)$. With these conditions in place, the dependence between $X_1,X_2,\nu$ may be arbitrary. For instance, we may suppose that $\nu$ is independent of $X_1,X_2$, or we may suppose that $\nu=X_1$, or ... .

Take any $c\notin\{1,2\}$ and let $Y_1,Y_2$ be obtained from $X_1,X_2$ by replacing $X_\nu$ by $c$, so that
\begin{equation} (Y_1,Y_2)=\left\{ \begin{aligned} (c,X_2) &\text{ if }\nu=1,\\ (X_1,c) &\text{ if }\nu=2. \end{aligned} \right. \end{equation} Then $$P(Y_1=c,Y_2=c)=0\ne p(1-p)=P(\nu=1)P(\nu=2)=P(Y_1=c)P(Y_2=c). $$ So, $Y_1,Y_2$ are not independent.