Convergence in distribution of the negative part of centered/scaled poisson variable

Question

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$

where $X_j \overset{d}{\sim} Poisson(1)$

Now split $T_n$ into its positive and negative parts: $T_n = T_n^{+} - T_n^{-}$, and let $F_n^{+}(t)$ and $F_n^{-}(t)$ denote its positive and negative distribution functions.

So I can see $F_n(t) \rightarrow^{CLT} \Phi(t)$, but what does $F_n^{-}(t)$ converge to? I would like to argue it converges to the negative part of a standard normal, but I am having trouble making the argument rigorous.

Answer 1

$F^+(t)=P(T\leq t|t>0)$ and $F^-(t)=P(T\leq t|t<0)$ Now, let $X_i=B_iX_i^+-(1-B_i)X_i^-$ where $B_i\sim Ber(0.5)$. Thus, the distribution of $X_i$ is just a mixture of $X^+$ and $X^-$, where the distribution of the two signed random variables are the conditional poisson distributions given $X\geq 0$ and $X<0$. Since each realization is either negative or non-negative, they represent disjoint regions on the range of $X$ and hence they contribute to the positive and negative areas of $F$