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Theorem 2 in Iain M. Johnstone. Zongming Ma. "Fast approach to the Tracy–Widom law at the edge of GOE and GUE." Ann. Appl. Probab. 22 (5) 1962 - 1988, October 2012. https://doi.org/10.1214/11-AAP819 …

You can deduce the result if it is already known that $X_1\to^d N(0,1)$. The convergence in law of $(X_1,...,X_k)$ boils down, for fixed reals $t_1,...,t_k$, to the convergence of the characteristic f …

$Y_i\sim^\text{iid} \text{Exp}(1)$ random variables, $f(Y)=\min_i Y_i$ has Exponential distribution with parameter $n$ with $f$ 1-Lipschitz. Then $E[f(Y)]=1/n$ but $$P(f(Y)-E[f(Y)]>t)=\exp(-n(t+1/n))$ …

The paper Chao, C., Bai, Z., & Liang, W. (1993). Asymptotic Normality for Oscillation of Permutation. Probability in the Engineering and Informational Sciences, 7(2), 227-235. doi:10.1017/S0269964800 …

At step $k$, we pick two indies $i\ne j\in [n]$ (say, wlog, $i$ is picked first, then $j$) and the probability that $i$ has been picked before at least once at steps $1,...,k-1$ is at most $$ \frac{2( …

If $Z\sim N(0,1)$ and $X$ is subgaussian with subgaussian norm less than $1/\sqrt 2$ then $$P(|X|>t)\le 2 e^{-t^2} \le K P(|Z|>t) $$ for some numerical constant $K>1$, thanks to lower bound on $P(Z>t) …

The distribution of $v^TB^{-1}AB^{-1}v$ is the same for every vector $v$ in the unit sphere either deterministic or independent of $W$. Once this is established, you are allowed to take $v=z/\|z\|$ in …

If $F(t)=1/n$, the Poisson/binomial approximation gives that $Y_n = nF_n(t)$ is Binomial($n,\frac1n$) and converges to Poisson(1), say in total variation distance. In particular, $P(Y_n=0)=P(F_n(t)=0) …

It is not possible to get $|\frac1n\sum_i z_i(w) - E[z_i(w)]|=O_P(r_n)$ with $r_n \lll n^{-1/2}$ even for a single $w$, since it would contradict the CLT whenever Var$[z_i(w)]>0$.