Simultaneous Concentration of $\sum_{i = 1}^{n} X_i^2$ and $\sum_{i = 1}^{n} X_i$ with $X_i$ iid. Poisson

Question

Consider $n$ independent Poisson(1)-distributed random variables $(X_i)_{1 \leq i \leq n}$.

This is a (hopefully more interesting) follow-up question to Super-exponential concentration for $\frac{\sum_{i = 1}^{n} X_i}{\sum_{i = 1}^{n} X_i^2}$ with $X_i$ iid. Poisson.

Is it true, that $$P\left(\sum_{i = 1}^{n} X_i^2 \geq cn \,\text{ and }\, \sum_{i = 1}^{n} X_i \leq c^{-1} n \right) \leq g(c, n) \cdot e^{-n}$$ for a (large) $c > 0$ and $g(c, n) \to 0$ for $n \to \infty$?

Answer 1

This is also not true. For instance the probability that $X_n^2 \geq cn$ is like $\exp(-\log n\sqrt{c n} )$. So in particular you can bound it below like \begin{align*} P(\sum_{i = 1}^n X_i^2 \geq cn, \sum_{i = 1}^n X_i \leq n/c) &\geq P(X_n = \lceil \sqrt{cn} + 1\rceil, \sum_{i = 1}^{n-1} X_i \leq n/(2c)) \\ &\geq e^{-o(n)} P(\sum_{i = 1}^{n-1} X_i \leq n/(2c)) \\ &\geq e^{-n(1 - 1/(2c))(1 + o(1))} P(\sum_{i = 1}^{n/(2c)} X_i \leq n/(2c)) \\ &=e^{-n(1 - 1/(2c))(1 + o(1))} \end{align*}

where the first inequality is from taking the event that $X_n \approx \sqrt{cn}$, the second is taking all but the first $n/(2c)$ to be $0$ and the last is via e.g. the central limit theorem. The point is that the first event is not too rare since the Poisson random variable is not subgaussian.