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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 paper by Heyde discussed in Iosif Pinelis' excellent answer concerns a more general situation where the summands are not stable but rather there is convergence to a stable law after normalization. …

The behavior of the order statistics is different than you predicted: Suppose that $k=n-\ell$, where $\ell=\ell(n)$ satisfies $\ell/n \to 0$. Denote by $Z$ a standard normal variable. Then for $b \in …

One possible definition is to assume a graph structure on the index set, where the maximal degree is $k$. Then you assume that every algebra $\mathcal{F}_i$ is independent of the join of the algebras …

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 …

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 …

Lyons, Russell, Robin Pemantle, and Yuval Peres. "Conceptual proofs of L log L criteria for mean behavior of branching processes." The Annals of Probability (1995): 1125-1138. https://www.jstor.org/s …

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 …

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. …

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 $ …

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? …