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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, …

Here is my counterexample: Take $f(x)=\|x\|^4$, then for any $x\neq 0$, by spherical symetry, $x$ is a eigenvector of $Hf|_x$. we have then $$Hf|_x(x)=a\|x\|^2 x $$ We can choose $x=v$ an eigenvector …

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 …

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 think the best way is to use the $L^2$ norm, because then exact calculation can be made in the Fourier space. $$\|f^{\otimes n}-g^{\otimes m} \|^2_{L^2(\mathbb{Z})}=\|\hat{f}^n-\hat{g}^m \|^2_{L^2([ …

Let $Y_i$ iid random exponential process, and $S=\sum_i Y_i$ then $(X_{k+1}-X_k)$ has the same random law as $$\frac{Y_k}{S} $$ For $n $ large, $Y_i $ are nearly independent. The maximum will then be …

I give here a solution at the limite $b$, $n$, $k$ very large: The optimal strategy is to alway play $\frac{b}{n}$ until you have more budget left than the number of head to get. It will give a probab …

I think you should have already a good estimate just comparing with the classical coupon colector's problem (CCP). Let us consider the CCP and pick the numbers one by one. We define the following sto …

We denote $\mu = (\frac{1}{n},\cdots,\frac{1}{n})$. First remark : because the convexity of $\ell^1$, for any $t$ the maximum of $\|\exp(tL)\nu-\mu\|_{\ell^1}$ is obtain when $\nu = \delta_x$, $x\in \ …

Call $X_k$ the amount of money at time $k$, $T_0:=\inf\{k,X_k=0\}$, $T_{n+m}:=\inf\{k,X_k=m+n\}$ and $T=T_{0}\wedge T_{n+m}$ First we have that $M_k:=X_k-(p-q)k$ is a martingale then (as already ment …

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{ …

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 …

I am proposing the following lemma. For $g$ concave $\inf_{|X|\leq a}\mathbb{E}(g(X-X'))=\inf_{|X|= p\delta_a+(1-p)\delta_{-a},p\in[0,1]}\mathbb{E}(g(X-X'))$ This show that for $a<2/7$, the solutio …

With a "physicist approach", I would write down the following equation for $f(x,t)$ that should represent the "density" of walker around $x$ at time $t$: $$\partial_t f =\Delta f -\alpha f^2 +\delta_0 …

What you want looks like very much the "Uniform Uncertainty Principle" in this paper of Candes and Tao https://statweb.stanford.edu/~candes/papers/OptimalRecovery.pdf which is proved for both Bernoul …