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Partial answer For every $a \in [0,1]$, $(\mu_n([0,a]))_{n \ge 0}$ is still a non-negative martingale, hence it converges almost surely to some random variable $L_a$ with values in $[0,+\infty]$. One …

The notations are contradictory. Once $p$ is fixed, and then it varies. Do you set $\mu_{n}:=\frac{1}{n}\sum_{i=0}^{n-1}T_{\ast}^{i}\nu$ for ALL $n \ge 1$ and assume that $T_{\ast}^{p} \nu = \nu$ for …

The answer to the first question is positive, and the lower bound is achieved, since the set of all probability measures on $[0,1]^k$ with uniform marginals is compact and since the integral of the bo …

Partial answer. Continuous (local) martingales are time-changed Brownian motion. A way to obtain funny local martingales which also are Markov processes is to start from a Brownian motion $B$ and its …

Is $Y$ independent of $(X_1,\ldots,X_n)$? If yes, write it. If yes, it suffices to prove that for any fixed unit vector $u$, $\langle u,Y \rangle \le c/\sqrt{d}$ with high probability. By rotational i …

An example: Matzinger studied the revocery of the environment from the second factor of a RWRE. https://arxiv.org/pdf/1110.6853.pdf https://matzi.math.gatech.edu/overview.pdf

The integral with regard do $\mathrm{d}M^*$ is a pathwise Stieltjès integral, so the question is an analysis problem. Let $f : \mathbb{R}_+ \to \mathbb{R}$ be any continuous function, $F$ its current …

NOT AND ANSWER, ONLY TOO WEAK INEQUALITIES AT THIS TIME You should replace all $n_i$ by $X_i^{(n)}$ in your question, and your proof. I did not read the proof, but the answer is trivially yes. Note th …

ADDENDUM. Vokram told me that I answered another question, not his question. Therefore, I give another counterexample (not so different from the previous one) disproving the equality. Choose a functio …

I use capital letters for random variables and small letters for possible values. Let $W$ be a brownian motion, defined on the canonical space $\mathcal{C}(\mathbb{R}_+)$ endowed with the Wiener measu …

A sufficient condition is unimodality of $f$, namely the existence of some $c \in~]a,b[$, such that $f$ non-decreasing on $[a,c]$ and $f$ non-increasing on $[a,c]$. If this property holds, then by Fat …

Let $\epsilon>0$ and $n \ge 1$. Then $$\bigcap_{k=1}^\infty\{|S_{n+k}-S_n|<\epsilon = \bigcap_{j \geq n}\{|S_j-S_n|<\epsilon\} \subset \bigcap_{j,k\geq n}\{|S_j-S_k|<2\epsilon\} .$$ Hence, taking comp …

Since the distribution of the r.v. $X$ is zero-mean, $X$ integrable (hence also the copy $Y$). The equality is attained iff the distribution of $X$ is carried by at most two points. Indeed, by indepen …

I call $f$ the density and $F$ the cumulative distribution function of $\mathcal{N}(0,1)$. Since $X_{(n)} \ge X_{(n-1)}$, $$X_{(n)}-X_{(n-1)} = \int_\mathbb{R} 1_{[X_{(n-1)} \le x < X_{(n)}]}dx.$$ Tak …

Partial answer The question may be whether the process $f$ is progressively measurable under the assumptions. If yes, we can conclude as follows. Fix $t \ge 0$. For each $\omega$, $f(\cdot,\omega)$ is …