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I agree with user39115! I will give a heuristic from random matrix theory because we know the global behaviour of the eigenvalue. First $$A=p 1 +\sqrt{N(p-p^2)}\frac{B}{\sqrt{N}} $$ where $1$ is the …

As Von Neumann said "Nobody really know what entropy is." so it is quite difficult to give a conceptual reason. However I think your calculation appears and can be interpreted in the statistical-mecha …

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 …

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, …

As proposed before, we can write $x_i$ as $$x_i=\frac{b_i}{b_1+\cdots+b_n} $$ where $b_i$ are independent exponential variable with parameter one. Let $y_i=e^{-b_i}$ then $y_i$ are iid uniform rand …

I have the following counter example: Let $I=Z_2\times G$ with $G$ a free groupe of rank $d$ and we will take $d$ large. $P_1$ is as follow $$P_1 \begin{cases}(1,w)\rightarrow (0,w):\forall w\in G, …

You should have a look on the book "Product of random matrices and application to Schrodinger operators" (Lacroix, Bougerol) http://fr.booksc.org/book/32773481/48bc02 or the paper of Le Page "Théorème …

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 …

There is a similar question here : Expected value of determinant of simple infinite random matrix I rewrite the heuristique from the random matrix theory: $$A=p^2 1 + (p-p^2)I_N+\sqrt{p^2(1-p^2)}\ti …

As Iosif Pinelis mentioned this is quite standard in large deviations theory so let me explain a bit the idea of the theorem he quote. Let $Y$ a random variable defined as $\mathbb{P}(Y=y)=\frac{1}{Z …

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 agree with michael, you should have more hypothesis for the CLT and as he says, one have to check that $\sigma^2 \neq 0 $. Another problem is that small covariance doesn't imply that the system is c …

The answers are yes and yes. Consider the Hermite Gauss functions : $\psi_n(x)=e^{-x^2/2}H_n(x) $, we have two properties: $$\psi_n''(x)+(2n+1-x^2)\psi_n(x)=0 $$ and $$\psi_{n+1}=\psi_n'(x) +x\psi_n …

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 \ …

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{ …