Search Results

We have $\phi(x)=\frac 1{\sqrt{2\pi}}\exp\left(-\frac{†^2}2\right)$ and $\Phi(x)=\int_{-\infty}^x\phi(t)dt$. We try to compute $$ I(a,b):=\int\phi(x)\Phi\left(\frac{x-b}a\right)dx.$$ Using the dominat …

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) …

The Kolmogorov distribution is defined by the distribution of the random variable $K:=\sup_{0\leqslant t\leqslant 1}|B(t)|$, where $B(t)$ is the Brownian Bridge. The problem of existence of moments fo …

For $\gamma\gt 0$ and a random variable $X$, define $$\lVert X\rVert_{\Phi_\gamma}:=\inf\left\{c\gt 0;\mathbb E\left[\exp\left(\left|X/c\right|^\gamma\right)\right] \leqslant 2\right\}.$$ For $\gamma …

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 …

Here are some ideas, which may be too long for a comment. Denote for fixed $n$ and $k$: $$ Z_{n,k}:=\sum_{i=1}^n\left(X_i\sum_{j=1}^k Y_{i+j-1 \mod n}-\frac 1k\right). $$ Let $$ A_{n,k}:=\sum_{i=1}^{ …

Let $U$ and $V$ be two i.i.d. random variables having finite expectation. Let $X_{3k}=X_{3k+1}:=U$, $X_{3k+2}:=V$ and $f\left(\left(x_i\right)_{i\in\mathbb Z}\right)=x_0x_1$. Then $\mathbb E\left[f\le …

Notice that $$\frac{S_{n-1}}{s_n}=\frac{S_{n-1}} {s_{n-1} }\frac{s_{n-1}} {s_n} $$ and by assumption, $\lim_{n\to +\infty}s_{n-1} /s_n=\sqrt{1-\rho^2} $, hence $S_{n-1} / s_n\to\math …

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 …

It is not exactly the mentioned result, but in Ouchti, Lahcen On the rate of convergence in the central limit theorem for martingale difference sequences, Ann. Inst. H. Poincaré Probab. Statist. 41 …