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Let $v\in \mathbb{R}^n$ be uniformly distributed on the unit sphere. Let $\lambda_1,...,\lambda_n$ be given real numbers. What is the distribution of $$X=\sum_{i=1}^n\lambda_iv_i^2\;?$$ Does it happen …

Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$? $$\mathbb{P}(X\geq x)$$ what is the order of the above probability as $x\to+\infty$?

What is the order of the following expectation with respect to $n$?: $$\mathbb{E}(\min_{1\leq i\leq n}|z_i|^2)$$ where $$(z_1,...,z_n)^T\sim N(0,I+11^T), 1=(1,1,...,1)^T$$ I know that when $z_i$ are …

Suppose $a_1,...,a_n\geq0, \sum_{i=1}^na_i=1$ and $Z_1,...,Z_n$ are i.i.d. standard normal, what is a sharp upper bound of the following probability as $\delta\to0$ and what is the order? $$\mathbb{P} …

Suppose $X_1,\ldots,X_n$ are i.i.d. standard normal. I'm wondering how to analyze the following quantity: $$\left|\frac{X_{(1)}X_{(n)}+X_{(2)}X_{(n-1)}+\cdots+X_{(n)}X_{(1)}}{n}\right|$$ where $X_{(1) …

Suppose a random positive definite matrix $A\in\mathbb{R}^{n\times n}$ has density function (with respect to the lebesgue measure on $\mathbb{R}^{n(n+1)/2}$) $f(A)=g(\lambda_1(A),...,\lambda_n(A))$ w …

Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\epsilon>0 …

Let $G(n,p)$ be an Erdos-Renyi random graph on $n$ vertices with probability $p$, i.e. for each pair of vertices, they are connected directly by an undirected edge with probability $p$. Suppose we are …

Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$. Suppose $\frac{A …

Let $X\sim N(\mu,\sigma^2I)\in \mathbb{R}^n$ be a non-centered ($\mu\neq 0$) Gaussian vector with independent coordinates. I'm wondering if there is any sharp lower bound of the following probability: …

Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, …

Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$- …

Can we solve the following recurrence relation: $$a_{m,n} = 1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}$$ with $a_{0,n}=a_{m,0}=0$? If not, can we get an estimate of the growth of $a_{m,n}?$ I encountered this …

I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant. Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ …

Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let $$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\ti …