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A good way to measure the set of maxima is the Hausdorff dimension of the set of records, which for BM is a.s. 1/2. Because of time/scale invariance, the dimension is the same for $\alpha B_\cdot$, an …

For $p>1$, the random variables you discuss do not possess exponential moments; You are in the regime of large deviations with stretched exponential tails. See for example the following recent paper b …

All you need is that the empirical measure of the $k_i$s converge. Whenever it does, just apply the original Pastur-Marcenko (1967) results to get that the empirical measure associated with your matr …

This is worked out in some detail in the paper of Fan, Grama and Liu, J. Math Anal. Appl. 448 (2017), 538-566 (see in particular Theorem 2.1 there, and the references). Unfortunately I do not have an …

Let $X$ be the array after your operation. The law of $X_{i,j}$ for $(i,j)$ chosen in a deterministic way from $[1,..,n]^2$ converges to the uniform law. The tilting due to your operations is very sma …

Your function is a continuous functional $F$ of the empirical process (up to an exponentially negligible error, it is the function $F(\mu)= \int\int h(x,y) \mu(dx)\mu(dy)$). Now apply the contraction …

There is a large literature, too long to review here. Deheuvels 1989 and Chistyakov 1996 come to mind. But also try: Google search and consider the relevant hits. Hope this helps.

I learnt the following from Govind Menon. Consider the Burgers equation $$\partial_t g+g \partial_x g=0$$ with initial condition $g(0,x)=1/x$. A self-similar solution is $g(t,x)=t^{-1/2}g_*(x\sqrt{t} …

What you wrote is incorrect - try the function $u(t,x)=x^2$ - it is of course not true that $W_t^2-1/2$ is a martingale when $W_t$ is standard Brownian motion..... You are missing an integral. You co …

This is essentially a product of local CLTs (since one can uncover the multinomial distribution variable by variable: decide first how many of the $n$ variables equal "1" with probability of success $ …

A general bound on the variance is given by the Borell (Tsirelson-Ibragimov-Sudakov) inequality, see http://webee.technion.ac.il/people/adler/borell.pdf Without more structure on A I don't think i …

It all depends on the shape of $P_2$ and on the assumptions you put on $X_i$. In what follows I'll assume that $\Lambda(\lambda)=\log E_1 e^{\lambda X_1}$ is finite for all $\lambda$. I will also ass …

The answer is yes. Let $F$ be the CDF of $X$, and let $G$ be the CDF of $Z_i$ (that is, $G(z)=P(Z_i\leq z)$). Then $F(x)= 1-(1-G(x))^n$, hence $F'(x)= nG'(x)(1-G(x))^{n-1}$. Thus, $$p_n(x)=\log F(x)' …

It is enough to understand the case $p/n\to 0$, as the eigenvalues of $XX^*$ match those of $X^*X$ up to an extra atom at $0$. In that case you can argue as follows, with $\sigma=1$: writing $W=n^{-1} …

This is a partial answer, in the regime that the probability in question is close to $1$. I normalize so that $EY_i=1$. The example $a_i=1/\sqrt{n}$ shows that you need to take $t\geq \sqrt{n}/2+O(1) …