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Let's try an easy case: $n=2$. If the matrix has entries $a,b,c,d$ we need $ab - cd$, $ac-bd$ and $ad-bc$ to all be nonzero. In particular if $b,c,d$ are all nonzero there are at most $3$ forbidden …

To restate the question in probabilistic language, each of the $n$ chldren's parents independently and with uniform probabilities chooses a $k$-element subset of $[n]$; say $X_i$ is the choice of chil …

If you're talking about the moments of a real-valued random variable, they are not independent in the sense that they are related by inequalities, e.g. $\mathbb E[X^2] \ge \mathbb E[X]^2$.

Why would you think normality is not needed? Consider the $1$-dimensional case: the left side is a constant times the variance of $x^2$, which depends on the $4$'th moment; it is not just a function …

The configuration space is the disjoint union of $\Lambda^N$ for each nonnegative integer $N$. You can take the Borel $\sigma$-algebra on each of these (or Lebesgue if you prefer, but you're unlikely …

Any probability measure $\mu_1$ absolutely continuous with respect to $\mu_1$ can be written as a Gibbs measure if you allow $G$ to take values $\pm \infty$. If the density is bounded above and below …

Yes: assuming $B_t$ is a random variable, $T$ is also a random variable, and the inf is done pointwise with respect to the sample space.

I'm assuming you mean $a, b, c, d$ to be independent exponential random variables with rate parameters $\lambda_1, \lambda_1, \lambda_2, \lambda_2$. I find that $(ac+bd)/(c+d)$ has mean $\lambda_1^{- …

Let $s$ be a computationally random bitstring. Consider $\tilde{s}$ defined by $\tilde{s}(2n) = \tilde{s}(2n+1) = s(n)$. Then $\tilde{s}$ should also be computationally random, but it does not contai …

If $P_k(t) = \sum_{m=0}^k a_{m,k-n} t^m$ is the generating function of an ascending antidiagonal, we have $$P_k(t) = \frac{t^k-t}{t-1} + \frac{1+t}{2} P_{k-1}(t), \ P_0(t) = 0 $$ and this can be solv …

It is possible to take random variables $X_k$ and the corresponding $W_k$ so $\mathbb E[X_k] \to \infty$ while $\mathbb E[W_k]/\mathbb E[X_k] \to 0$. For example, consider $X = N$ with probability …

Your two estimators $c_{XY}/n$ and $c_{XZ}/n$ are unbiased estimators of $P(X=Y)$ and $P(X=Z)$ respectively. They are not independent, however. Whether that is "acceptable" might depend on what you' …

Let $X = \sum_{i=1}^n a_i Z_i^2$. If $m = \min(a_1,\ldots,a_n)$ and $M = \max(a_1,\ldots,a_n)$, we have $m A \le X \le M A$ where $A$ has $\chi^2$ distribution with $n$ degrees of freedom. Thus $$\ …

1) To see that $K^{1/2}(H)$ is dense in $H$: if not, there is some nonzero $v$ orthogonal to it. But since $K^{1/2}$ is self-adjoint, that says $0 = (K^{1/2})^* v = K^{1/2} v$, and then $K v = K^{1/2 …

I presume you mean $\epsilon = \epsilon_i$, where $\epsilon_i \sim \mathscr N(0,1)$ is independent of all the previous random variables. The pairs $(\theta_i, \hat{\theta}_i)$ form a Gaussian Markov …