I'm thinking about the following question:


If $X\_n$ and $X\_n/Y\_n$ both converge in distribution towards a standard Gaussian random variable and $Y\_n \geq 0$ for all $n$, does then $Y\_n$ necessarily have to converge in distribution towards the constant $1$?


Does it change anything if we assume that all $X\_n$ have densities?


I've tried to reformulate things in terms of characteristic functions, assumed the existence of densities as stated above and tried to attack the problem using distribution theory (after a logarithmic transform, the above translates into an equation involving density convolutions) but didn't have any success. 


Any help is highly appreciated!



Does $X_n \xrightarrow{d} N(0,1)$ and $X_n/Y_n \xrightarrow{d} N(0,1)$ imply that $Y_n \xrightarrow{d} 1$?
Sorry, I don't have the reputation to comment, or I would've written something a bit shorter.


Briefly: I don't think that anything like this can be true without substantial assumptions beyond smoothness (e.g. maybe something like this is true if $X\_{n}$, $Y\_{n}$ are independent). To see this, let's just assume that $X\_{n}$ and $\frac{X\_{n}}{Y\_{n}}$ have exactly normal distribution, and fix two intervals $I\_{1} = [a\_{1},b\_{1}]$ and $I\_{2} = [a\_{2},b\_{2}]$ with the same mass under the normal distribution and with $0 \leq a\_{1} < b\_{1} < a\_{2} < b\_{2}$. We then define $Y\_{n} = 1$ when $X\_{n} \notin I\_{1} \cup I\_{2}$. Conditional on $X\_{n} \in I\_{1}$, we construct $Y\_{n}$ so that $\frac{X\_{n}}{Y\_{n}}$ has the $N(0,1)$ distribution conditioned on being in $I\_{2}$ and so that $Y\_{n}$ is monotone in $X\_{n}$ within this interval. Similarly, if $X\_{n} \in I\_{2}$, we construct $Y\_{n}$ so that $\frac{X\_{n}}{Y\_{n}}$ has the $N(0,1)$ distribution conditioned on being in $I\_{1}$ and so that again $Y\_{n}$ is monotone in $X\_{n}$. In other words, we use $Y\_{n}$ to `swap' two intervals. It shouldn't be so hard to write down exactly what these conditional distributions of the $Y\_{i}$'s look like in terms of the CDF's of the conditioned normal distribution, and everything has smooth densities and inverses and so on. In this construction, $Y\_{n}$ is not always close to 1, especially if $a\_{2},b\_{1}$ are well-separated - $Y\_{n}$ is small when $X\_{n} \in I\_{1}$ and large when $X\_{n} \in I\_{2}$.


