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Suppose the random variables $X \geq 0$ and $Y \geq 0$ are both stochastically dominated by $Z \geq 0$, i.e. \begin{align*} & P(X \leq x), P(Y \leq x) \geq P(Z \leq x) \ , \ \forall x \geq 0 \ . \end{ …

Let $\lambda_1(A_n),...,\lambda_n(A_n)$ be the random eigenvalues of a random $(n \times n)$ matrix $A_n$. We can define the empirical spectral measure $\mu_n^{A_n}$ on $(\mathbb{C},\mathcal{B}(\mathb …

It's been a while since the question was asked, but I randomly came across this article by M. Ledoux, which seems to give a five term recurrence relation. See Theorem 2 of: http://www.numdam.org/artic …

Let $S_{d-1}$ denote the unit sphere in $\mathbb{R}^d$ and let $(Z_x)_{x \in S_{d-1}}$ be a gaussian process with mean zero and covariance structure given by the square of the scalar product, i.e. $$ …

For fixed $k$ suppose we have $X_1,\ldots,X_k$ non-negative random variables with density functions. Setting a): We know $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$ exactly for any integers $n,m \in \mathbb{ …

Let's assume $\mathbb{E}[X_i]=0_{\mathbb{R}^d}$, since PCA makes little sense otherwise. (Always normalize your data points by subtracting the empirical mean.) I'm afraid your assumption, that $\lambd …

Using the Radon-Nikodym Theorem this problem can be reduced to the case for a fixed measure. Suppose none of the $\mu_n$ is the zero-measure, i.e. $\mu_n(X)>0$ for all $n \in \mathbb{N}$. For a series …