If we assume WLOG that our variable X has mean zero (mean-centered), then this can be stated
$Pr \bigg(\sum x^2 > \sum (x-n)^2 \bigg)$
for some random variable $n$ distributed under $N \sim N(0, \sigma_N^2) $.
Expanding the quadratic expression and arranging terms gives
$Pr \bigg( \sum (2 x n ) > \sum ( n^2 ) \bigg)$
Under what conditions is it possible to derive this probability? In this case, I know from domain context that var(X) > var(N), but var(N) is not negligible.
This feels strange because, intuitively, adding noise should be far more likely to increase variance, given the nature of squared error terms. Looking at this equation, the LHS has expected value = 0 due to independence of X and N, while the RHS has expected value = $\sigma_N^2 \cdot n_{obs}$. So the greater $\sigma_N^2$ becomes, the less likely this value, but I'm wondering how to bound or quantify this changing probability.
I'm interested to at least see what can be derived by assuming a Gaussian distribution for X, but I'm more interested to see if any generalities (e.g. lower or upper bounds) can be made overall.
Given that we're dealing with summations, is it reasonable to invoke the CLT on the LHS, and claim that var(XN) = var(X)var(N) given that X and N both have mean zero? What are the cautions against invoking the CLT in this case?