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Yes, there are such inequalities, which follow from the Kac Lemma in Ergodic Theory. (Although the form of the inequality at the end of the post is too optimistic- did you want the supremum to be tak …

For $0<\beta \le 1/2$, any limiting distribution of the rescaled process $S_n/n^\beta$ will be fully supported in $[-1,1]$, so it will not be normal. The downward drift will imply (using Hoeffding's …

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? …

Since the continuous image of a compact set is compact, it suffices to determine whether the mapping $p \to F(x,p)$ is continuous. This is the case for most natural parametrized families of distributi …

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 …

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 …

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 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 …

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

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 …

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 …

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 …

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 $f$ is the sum and all the components are identical then the inequality fails for $t=n$.

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. …