MathOverflow

I am reading this paper: https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-040120-022239
Right before Remark 3, the authors state "Consider the assumption of Theorem 1 with $B_n^2\log(pn)^5 = o(n)$.Then we have $$\max_{j=1,2\dots,p}\big\vert\sigma_j^2/\hat\sigma^2_j - 1\big\vert = o_P\!\left(\log(p_n)^2\right)$$ [...]". Here $\sigma_j^2...

We know that for ordinary homotopy types we have several formulations of rationalizations, e.g. Bousfield-Kan’s $\mathbb{Q}$-completion, Sullivan’s rationalization, Bousfield’s homology rationalization, Casacuberta-Peschke’s $\Omega$-rationalization...

I am trying to interprete the discussion given in Section 3 of this paper,https://core.ac.uk/download/pdf/82217936.pdf
Lets suppose we restrict to considering Gibbs's measures of the form $\sim e^{-V}$.
Then is there any easy sufficient condition on $V$ which ensures that this Gibbs' measure is weak-PI ?...

Let $n$ and $k$ be positive integers. Let $X$ be the empirical mean of $n$ iid Rademacher random variables. Note that the distribution of $X$ is symmetric about 0, and also $|X| \le 1$ w.p 1. Let $X_1,X_2,\ldots,X_k$ be iid copies of $X$. Consider the...