I found this question:https://mathoverflow.net/questions/420837/chernoff-style-concentration-bound-for-ratio-of-variables.
I want to ask if we get similar thing for the ratio of the sum and the one Gaussian variable.

Given i.i.d. Gaussian random variables $X_1,\dots, X_k$ with $N(0, 1)$. Fix $\epsilon\in (0,1)$, can prove that for any $\delta>0$ there exists $1\le k=k(\epsilon, \delta)$ (some number) so that
$$ 
P\left(\frac{X_1^2+X_2^2+\dots+X_k^2}{X_1^2}>\frac{1}{\epsilon^2}\right)>1- \delta ?
$$

Can we find such $k$?


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In this slide: https://nobel.web.unc.edu/wp-content/uploads/sites/13591/2020/10/Probability_Inequalities.pdf,
for $Y\sim \chi_k^2$ (Y=\sum_{I=1}^k X_i^2), for $t\in (0,1)$
$$
P(Y\ge (1+\epsilon)k)\le \exp(-k(t^2-t^3)/4).
$$