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We can write $h(x)=\frac 1{\sqrt{2\pi}}\int_{-\infty}^x \exp\left(\frac{x^2-y^2}2\right)dy$. Now put $t=x-y$. We get \begin{align} h(x)&=\frac 1{\sqrt{2\pi}}\int_0^{+\infty}\exp\left(\frac{x^2-(x-t) …

We actually have that $\mu=\nu$ is guaranteed if and only if $\sum_{n\in S}\frac 1n$ is divergent. It's a condition which translates the fact that the set of indexed $k$ such that $\mu_k=\nu_k$ has to …

No, even in the most favorable case $(X_i)_{i\geqslant 0}$ iid with $\mathbb P(X_i=1)=\mathbb P(X_i=-1)=1/2$. Denoting $F_n$ the cumulative distribution function of $n^{-1/2}S_n$, we have by symmetry …

One can use Birnbaum and Marshall inequality: Theorem(Theorem 2.1. in 1). If $\left(S_k,k\geqslant 1\right)$ is a non-negative sub-martingale and $(c_k,k\geqslant 1)$ a non-decreasing sequence of pos …

We know by Ulam's theorem that a Borel measure on a Polish space is necessarily tight. If we just assume that the metric space is separable, we have that each Borel probability measure on $X$ is tight …

One of the most recent results in this area is Katarzyna Bartkiewicz, Adam Jakubowski, Thomas Mikosch, Olivier Wintenberger, Stable limits for sums of dependent infinite variance random variables, P …

As Christian Remgling's example $\mu_n:=\delta_{e_n}$ shows, the convergence of the characteristic function of $\mu_n$ to some characteristic function does not even guarantee tightness. It's worth poi …

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following: Let $(X_n,n\in\Bbb N)$ be a strictly stationary $\phi$-mixing sequence, f …

Assume that the random variables $X$ and $Y$ are defined on the probability space $(\Omega,\mathcal F,\mu)$. Let $\Delta:=\{(x,y)\in\Bbb R^2,x\lt y\}$. We have by independence $$ E\left[e^{it\max(X,Y) …

Consider an independent sequence of events $\left(A_i\right)_{i\geqslant 1}$ such that if $2^N+1\leqslant i\leqslant 2^{N+1}$, $\mathbb P(A_i)=2^{-N}$. Define for $2^N+1\leqslant i\leqslant 2^{N+1}$ t …

Let us state Corollary 2.1 of these notes. Let $p>1$, $X\in\mathbb L^p$ and let $\left(\mathcal F_n\right)_{n\geqslant 1}$ be a filtration. Denote by $\mathcal F$ the $\sigma$-algebra generated by …

I think the statement of Theorem 16.3 Billingsley meant is Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, $(\xi_i,i\geqslant 1)$ be i.i.d. zero mean random variables (for $\mathbb P$ …

Let $(\Omega,\mathcal B,\mu)$ be a probability space and $\mathcal A$ a sub-sigma-algebra of $\mathcal B$. The following statements are equivalent: $\mathcal A$ admits a countable set of generators …

The following is contained in Durrett's book Probability theory and examples. First remark: the limit always exists. Consider indeed $U_n$ the number of upcrossings of $[a,b]$ by $M_0,\dots,M_n$. The …

If $X$ is an integrable random variable and $\left(\mathcal F_n\right)_{n\geqslant 1}$ is a filtration, then $X_n:=\mathbb E\left[X\mid\mathcal F_n\right]$ is a martingale. It is worth mentioning tha …