For a standard normal vector $P\{\|x\|_2^2 > n \theta\}$ is monotonic in $n$

Question

I want to prove that for standard normal vectors $x \in \mathbb{R}^n, x \sim \mathcal{N}(0, I_n), x' \in \mathbb{R}^{n + 1}, x' \sim \mathcal{N}(0, I_{n + 1})$: $$P\{\|x\|_2^2 > n \theta\} \geqslant P\{\|x'\|_2^2 > (n + 1) \theta\}.$$

This fact is used in the article in the proof of Theorem 2.3, however, no explanation or citation is given. Is it obvious?

It seems that $\frac{\|x\|_2^2}{n}$ has gamma distribution, but writing down the explicit formulae for the probability did not get me far.

UPDATE: From the answer of @MMH and from my experiments it seems that for each n there exists a $\theta_n$ after which the inequality is true, and this $\theta_n$ gets closer to 1 with the increasing n. So probably the question now is to find the formula for $\theta_n$ and to understand why for large $\theta$ this is correct.

Answer 1

Consider the simple case $n=1$. Then, we can plot $\mathrm{Pr}(x_1^2 \geq \theta)$ and $\mathrm{Pr}(x_1^2 + x_2^2\geq 2\theta)$ and verify the claim. Note that using scipy package we can plot these two probabilities (see here). I wrote a simple code to verify the claim:

import numpy as np
from scipy.stats import chi2
import matplotlib.pyplot as plt

Theta = np.linspace(0,4,1000)
y1 = [1-chi2.cdf(x=theta, df=1) for theta in Theta]
y2 = [1-chi2.cdf(x=2*theta, df=2) for theta in Theta]
plt.plot(Theta,y1,'-r',label='n=1')
plt.plot(Theta,y2,'-g',label='n=2')
plt.xlabel('theta')
plt.legend()
plt.ylabel('tail probability')
plt.show()

The output is:

It seems the claim is not valid for every $\theta\geq 0$ and $n \in \mathbb{N}$.