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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
4
votes
1
answer
296
views
Rank of a sequence of covariance matrices
Let $X_i$ ($i=1, \dots$) be an orthonormal basis for $L^2(\Omega, \mathbb P)$. In particular, it holds that
$$\mathbb E[X_iX_j] = \delta_{ij}.$$
Now take $Z\in L^2(\Omega, \mathbb P)$ and define $\ti …
0
votes
1
answer
4k
views
Convergence of the empirical distribution function
Let $\alpha\in\mathbb R^d$, with $\alpha\neq 0$. Take a sequence of iid random variables $X^{1},\dots,X^{n}$ with values in $\mathbb R^d$ and denote $R$ the cdf of $\alpha X^1$ (where the product is a …
4
votes
1
answer
1k
views
Quantile convergence
Let $X^1,\dots,X^n$ be a sample of (not necessarily iid) random variables and denote
$$F^n(x)=\frac{1}{n}\sum_{i=1}^n \mathbf 1_{X^i\leq x}$$
the empirical distribution function. Suppose that we know …
0
votes
Accepted
Estimate on gaussian distribution
Based on Carlo's contribution, after short manipulations I got to the answer
$$f(M)=\left(1-\exp\left(-\frac{M^2}{d^2 \|C\|^2}\right)\right)^d,$$
for the full rank case, where $\|C\|=\max |C_{ij}|$ an …
4
votes
2
answers
325
views
Estimate on gaussian distribution
Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$
and such that …