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For $np\rightarrow\infty$, it is a very classical result that it converges to the semi-circle law. If you just want the symmetry in a general case, I suppose one should just estimate $$\frac{1}{n}\mat …

Let $(X_i)_{i\in \mathbb{N}}$ iid random variables such that there exists $\alpha>0$ where $\mathbb{P}\left(X_1\in [x,x+1]\right)\leq \alpha$ for all $x\in \mathbb{R}$. Assume $\alpha$ small enough, …

You can write $$X_t-X_t'=x-x'+\int_0^t(\mu(s,X_s)-\mu(s,X_s'))ds+\int_0^t(\sigma(s,X_s)-\sigma(s,X_s'))dW_s $$ Then $$ \mathbb{E}(\|X_t-X_t'\|^2) \leq 3 \left(|x-x'|^2+a^2t\int_0^t \mathbb{E}(\|X_s-X_ …

Can't you just sample a point $p$ uniformly on $\mathcal{S}$. Then draw a independent $u$ variable uniformly on $[0,2]$. If $u\leq\Delta(X,p)$ keep $p$ and you are done. If $u>\Delta(X,p)$ then forget …

You can modify a little the problem replacing the deterministe $t$ by a random Poisson variable $\hat{t}$ of expectation $t$. For large $t$, because $|\hat{t}-t|\lesssim\sqrt{t}$ (Central Limit Theore …

We denote $L_{\epsilon}(x):=\{(P_{1}+\epsilon_{1})x,\cdots,(P_{m}+\epsilon_{m})x\text{ linearly independant}\}$ First we have that for any fixed $x\in\mathbb{R}^{n}$, $$\mathbb{P}(L_{\epsilon}(x))=0.$ …

I start with this simple remark: the tridiagonal matrix $$A_k=\begin{pmatrix}0 & 1 & & & \\ 1 & 0 & 1 & & \\ & 1 & 0 & \ddots & \\ & & \ddots & & 1 \\ & & & 1 & 0\end{pmatrix}$$, $A_k\in \mathbb{R …

I would use a "Girsanov like" trick to modify the expectation: For a general random walk with $\mathbb{P}(X=1)=p$ that we denote the associate law $\mathbb{E}_p$.For $\alpha\in \mathbb{R}$ we have $$\ …

If we assume $A$ constant $$\frac{d}{dt}\mathbb{E}(X_t )=A \mathbb{E}(X_t ) $$so $\mathbb{E}(X_t)=e^{tA}X_0$ and $\mathbb{E}Y_t = Y_0 + \int_0^t H_s e^{sA}X_0ds$. For the variance, we can assume $X_0= …

$$\mathbb{P}(B_{n+t,p}=B_{n-t,p})=\sum_{k=0}^{n-t}\frac{(n+t)!(n-t)!}{(n+t-k)!(n-t-k)!(k!)^2}p^{2k}(1-p)^{2n-2k}. $$ We write $a_k$ the terms in this sum. We have $$ \frac{a_{k+1}}{a_k}=\frac{p^2(n+t- …

I consider the case of discrete probability. There exists $n\in \mathbb{N}$, $A_1,\cdots,A_n$ and $p_1,\cdots, p_n>0$ such that $\sum_{i=1}^n p_i = 1$ and $\mathbb{P}(A=A_i)=p_i$. Then we denote $X_1, …

I can propose a simple heuristic but it doesn't go as far as Angela Zhou conjecture. For each face the probability that it appears $k$ times in $N$ toss is $\frac{N!}{k!(N-k)!}\frac{(N-1)^{n-k}}{N^N} …

Let $\Omega$ a finite metric space with $\forall x,y,z\in \Omega:d(x,y)<d(x,z)+d(z,y)$. Does there exists a continuous-time Markov process $X$ on $\Omega$ such that $$\mathbb{E}_x(T_y)=d(x,y)$$ for al …

This seems to me a standard exercice in a probability course: Ignore the $1/(t+1)$ term as $X(t)\geq e^{-1}\exp(W(t)-t/2)$. The term $M_t:=\exp(W(t)-t/2)$ is well known to be a martingale so $\mathbb{ …

What you want for 2- seems to me similar to a large deviation result (https://en.wikipedia.org/wiki/Large_deviations_theory) of speed $n^2$ for the diagonal term $$\frac{1}{n^2}\log\mathbb{P}(\sum_i …