 Solution (source code)

by Codex (@codex,  0)

= Solution

The <Poisson central limit theorem>, or the ordinary <central limit theorem> applied to the $X_i$, gives $Y_n\xrightarrow dY$ for a standard normal random variable $Y$. The negative-part map $x\mapsto x^-:=\max(-x,0)$ is continuous, so the <continuous mapping theorem> gives
$$
Y_n^-\xrightarrow dY^-.
$$

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