Asymptotic independence characterization in the two envelopes problem

Let $X$ be a positive random variable, and write

$$\log_2 X=\lfloor\log_2 X\rfloor+\{\log_2 X\},$$

where $\lfloor\log_2 X\rfloor$ and $\{\log_2 X\}$ are respectively the whole and fractional parts of $\log_2 X$. Let $A$ and $\Delta$ denote the corresponding variables in the two envelopes problem. Asymptotic independence characterization. $A$ and $\Delta$ are asymptotically independent if and only if the fractional and whole parts of $\log_2 X$ are asymptotically independent, with the whole part asymptotically uniformly distributed over all integers. This characterizes the proposed asymptotic-independence behavior in the two envelopes problem; the surrounding results establish partial implications involving weak convergence and total variation, but the conjectural characterization and the appropriate notions of approximation remain to be established.