If I have sequences of random variables $X_n$, $Y_n$ with $X_n$ known and observable but $Y_n$ not, and I know that ${X_n}/{Y_n} \xrightarrow{d} N(0,1)$. Under what conditions is $Y_n$ identifiable simply from this convergence? it seems to me like it should not be unique without quite strong assumptions but not sure if there are any results in this area, and perhaps it depends on the distibution of $X_n$?

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    $\begingroup$ Since convergence (in any sense) of a sequence does not change when any finite number of terms are removed, it's difficult to see how one could deduce anything at all about any particular $Y_n$ (or, indeed, any finite number of them). Are you sure you have formulated your question as you intended? $\endgroup$ – whuber Jul 30 at 14:06