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The random variable $X$ is distributed like the total progeny of a critical branching process $(Z_n)$ with binomial $(d,\frac1d)$ reproduction distribution, starting from $Z_0=1$. As such, indeed $E(X …

The transformation $L=TG$ is defined on vectors $x$ with positive coordinates by $$ Lx(s)=\sum_uq(u|s)\mathrm{e}^{-r(u)}Mx(u),\quad\mbox{where}\ Mx(s)=\prod_ux(u)^{p(u|s)}. $$ Thus $M$ and $L$ are ho …

A partial answer is that, when $X$ is not integrable, $\hat X_x$ the residual lifetime of $\min\{X,x\}$ converges to infinity in distribution when $x\to\infty$. Since the residual lifetime $\hat X$ of …

To know the total number of fishes in the pond at time $t$, one needs to know the lifetimes of the $N$ initial fishes. The number $N_t$ of fishes in the pond at time $t$ not in the pond at time $0$ is …

Nicolas Th. Varopoulos, Brownian motion and random walks on manifolds, Annales de l'Institut Fourier 34(2) (1984), 243-269. Abstract: We develop a procedure that allows us to “discretise” the Brownia …

(This is to answer a question about Poisson processes asked by the OP in the comments.) Here is an analog of Alekk's argument about the mutual singularity of the probability distributions of diffusio …

For every $t$, let $Y_t=\log(Z_t^2)$. Fix some $t$. The sequence $(Y_{t-k})_{k\geqslant0}$ is i.i.d. with $E[Y_t]\lt0$ hence the usual law of large numbers yields $\frac1j\sum\limits_{k=0}^{j-1}Y_{t-k …

You seem to be after $p(x,t)=P[S_t=+]p_+(x,t)+P[S_t=-]p_-(x,t)$ where $S_t$ is the state at time $t$. One knows that $S_0=+$ and that $S_t$ switches from $\pm$ to $\mp$ at rate $\kappa_\pm$. Thus, $ …

Assume without loss of generality that each $X_i$ is standard normal and that $Y$ is normal with mean $\mu$ and variance $\sigma^2$. By definition, for every $y$, $$ P[y\gt\max(X_1,\ldots,X_N)]=\Phi(y …

The horror, the horror... :-) Recall that $\Phi(x)=P[X\leqslant x]$ for every $x$, where the random variable $X$ is standard normal, and that, for every suitable function $u$, $$ \int_{-\infty}^{+\in …

More generally, let $A$ denote some subset of the state space of a homogenous Markov process $(Z_t)_{0\leqslant t\leqslant t_0}$ with transition kernel $(a_{t-s})_{0\leqslant s\leqslant t\leqslant t_0 …

The inequality holds for every $n\ge2$ (integer or not) and every probability distribution. Here is a proof. We begin with two easy facts. Fact 1: For every $z\ge0$ and every $k\ge1$, $$ zF(z)^k-\in …

To specify the probability space is rarely a good idea but since you insist on it, a possible choice would be $\Omega=K^\mathbb N$, where $K=\{1,2,\ldots,k\}$ denotes the set of types of coupons, endo …

The usual infinitesimal analysis leading to a differential system applies, with a twist: for small $s>0$, $$ p_{i0}(k,t+s)=(1-\lambda s)p_{i0}(k,t)+\mu sp_{i1}(k-1,t)+o(s), $$ and $$ p_{i1}(k,t+s)= …

Assume the distribution of $X_n$ is binomial $(n,p)$. Then, for every real number $t$, $$ \mathrm E(\mathrm e^{\mathrm it(X_n-np)})=\mathrm e^{\mathrm itnp}(1-p+p\mathrm e^{\mathrm it})^n. $$ Assuming …