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One knows that $P(S_n,n)$ is a martingale if and only if $P(s+1,n+1)+P(s-1,n+1)=2P(s,n)$ and that $Q(B_t,t)$ is a martingale if and only if $2\partial_tQ(x,t)+\partial^2_{xx}Q(x,t)=0$. Assume that $P …

The result holds if, additionnally to the conditions of the post, one assumes that the vector $(X,Y)$ is Gaussian. Then, $Y=aX+\sqrt{1-a^2}Z$ with $a=\mathrm{cov}(X,Y)$ and $Z$ standard Gaussian such …

Let $M_n(T)=\max\{S_1(T),\ldots,S_n(T)\}$ where the $S_j(T)$ are i.i.d. and distributed like $S(T)$. A partial answer to your question is that $E(M_n(T))/T\to\mu$ when $T\to+\infty$ as soon as the f …

Tom is right: the proof of Chebyshev's inequality can be easily adapted to every nondecreasing nonnegative function. The proof of this generalization that I prefer has a principle worth remembering: …

You are looking for random variables $X$ of median $m(X)$ such that $$ E((X-E(X))^3)>0\quad\mbox{and}\quad E(X)< m(X). $$ Assume there exists two nonnegative random variables $Y$ and $Z$ and an indep …

A mathematical reason is as follows. On the one hand, the Laurent series for the modified Bessel functions of the first kind $I_k$ can be deduced from the Laurent series for the Bessel functions of t …

The exponential generating function of the sequence $(f(n))$ is $$ \sum_{n\ge0}f(n)\frac{s^n}{n!}=\mathrm{e}^{s}\sum_{k\ge0}(1-\mathrm{e}^{-s/2^k}). $$ Not sure that this formula helps to recover the …

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 …

If $(x_n)$ solves the recursion you are interested in, then the sequence $(y_n)$ of general term $y_n=x_n/\alpha^n$ is a random Fibonacci sequence such as in the Embree-Trefethen page you linked to.

Indeed, these formulas are standard. Their derivation in a slighly more general setting than yours is as follows. For every $t\ge0$, let $$ M_t=\displaystyle\exp\left(\int_0^tv(X_s)\mathrm{d}s\righ …

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 …

The trouble (as was already explained to you) lies in the starting point $t=0$ of the integral in the exponential. Fortunately, W+W are only interested in steady solutions of equation (15) of their ar …

You ask for the correlation function, defined for every $0\le s\le t\le u\le v$ by the formula $$ C(s,t;u,v)=E(Y_{s,t}Y_{u,v})-E(Y_{s,t})E(Y_{u,v}). $$ One first computes $E(Y_{s,t})$. For every $t\ge …

This is a general fact: assume that $X$ is a positive random variable and that, for a given nonnegative function $g$, $\displaystyle E(\mathrm{e}^{-y/X})=\int_y^{\infty}g(x)\mathrm{d}x$ for every posi …

Call $[A_n,A_n+V_n]$ the segment at time $n$ assuming that $A_n$ is the end of the segment around which it rotates between times $n$ and $n+1$. Thus $A_n$ and $V_n$ are complex numbers, $A_0=1=V_0$, a …