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Given $n^2$ i.i.d. uniform points in $[0,1]^2$, the goal is to draw a configuration of $cn$ vertical lines and $cn$ horizontal lines such that in each small rectangle there is at most one marked point …

If the random vector $(X,Y)$ in the plane has independent coordinates and a rotation-invariant distribution, then it is Gaussian.

The length $R_N$ of the longest run in the first $N$ digits satisfies $R_N/\log_b(N) \to 1$ almost surely as $N \to \infty$, as first proved by Renyi, see the discussion in [1]. (Many references focus …

Let $A:=\Bigg\{\bigg\vert\dfrac{\sum_{j=1}^n(\sum_{i=1}^n a_{ij})^2}{n^2} -1\bigg\vert > \epsilon\Bigg\}\,$ denote the event in question. We will show that $\operatorname{P}(A)\le C_{\epsilon}/n$ f …

There is$^*$ a counterexample in the atomic case, see below, so we will assume that $(\Omega, \mathrm{P})$ is a non-atomic standard Lebesgue probability space (so it is Isomorphic to the unit interval …

Shishir, this is quite elementary: write the probability $P_2(\omega)$ of any individual bit sequence $\omega$ as $P_1(\omega) f(\omega)$ where by definition, $f(\omega)=\exp(-\hat{kl}_n)$. Finally, s …

Perhaps most relevant is [1] which discusses this construction and reduces the general case to $c=2$. See Theorem 1 and proposition 15 there. Note that $cp \le 1$ is not a sufficient condition; an in …

The hypothesis implies that $M_k=[e^{\sum_{n=1}^k X_n}]$ is a supermartingale, with $M_0=1$. Then the optional stopping theorem for positive supermartingales implies the requested inequality. (see e. …

Such a sequence $a_n$ does not exist even for a well studied example like returns to the origin of simple random walk in one dimension. If $X_i$ denotes the number of steps from the $i-1$ time the wal …

The implication goes the other way. The "standard" inequality you quote, usually called McDiarmid's inequality is derived from the second inequality that you ask about. See e.g. http://empslocal.ex.a …

Assume $X>0$ a.s. (so the constraint can be satisfied) and write $Y=\log X$. By Jensen's inequality (https://en.wikipedia.org/wiki/Jensen%27s_inequality), $\mathbb{E} X \ge e^{\mathbb{E} Y}=e^C$, so …

For Markov chains, a very useful condition is Harris recurrence, see https://en.wikipedia.org/wiki/Harris_chain. This has been generalized to continuous time, see https://www.jstor.org/stable/3690386? …

This Theorem is a direct consequence of the Skorohod representation of Martingales. You can find it, along with many variants, in Hall, Peter, and Christopher C. Heyde. Martingale limit theory and its …

As Iosif wrote, the conjecture does not hold. Suppose $\alpha=1$ and $X_1$ takes the values 0,1,2 with equal probability and $X_2$ takes the values 0,2 with equal probability. Then $h(1)=0$ but transl …

Let $J$ take the value $M$ with probability $1/M^2$ and the value 1 with probability $1-1/M^2$. Let $R_i$ be i.i.d. $\pm 1$ valued random variables of of mean zero, and define $X_i:=JR_i$. Then for $ …