Search Results

They are not equivalent. Suppose $X = [0,1]$, $\mu$ is a unit mass at 0, and $x_n = 1/n$. This sequence is $\mu$-uniformly-distributed-B, because for any continuous $f$, $\int f(x) \, d\mu = f(0) = …

Presumably your Gumbel random variables are independent. A Gumbel random variable $X_i$ with location parameter $\alpha_i$ and scale parameter $\beta_i$ has CDF $$ F_i(x) = \exp(-\exp((\alpha_i - x)/\ …

Using $\max(a,b) = \dfrac{a+b}{2} + \left| \dfrac{a-b}{2}\right|$, write $u = w_1 + |w_2| - |w_3|$ where $$ \eqalign{ w_1 &= C (y_1 + y_2) \cr w_2 &= \dfrac{1}{2} (x_2 - x_1 + C (y_2 - …

For a normal distribution with mean $0$ and standard deviation $\sigma$, I get (with help from Maple) $$W_X(t) = \sigma^2 + \sqrt{\frac{2}{\pi}} \frac{\sigma t}{1 - \text{erf}(t/(\sqrt{2}\sigma))} e^{ …

Using the Poisson summation formula, I find that the variance is $$ \sigma^2 + \dfrac{1}{12} + \sum_{k=1}^\infty (-1)^k e^{-2\sigma^2 k^2 \pi^2} (4 \sigma^2 + 1/(\pi^2 k^2)) $$ If $\sigma$ is not t …

Presumably you mean you have continuum-many independent Brownian motions, one (call it $W_p(t)$) with $W_p(0) = p$ for each $p$. Unfortunately I'm pretty sure the number of $p$ for which $W_p(t) = 0$ …

Generalizing Efron, we can get probability $\ge 70\%$ with six 10-sided dice: 10 sides $=6$ 3 sides $=11$, 7 sides $=5$ 4 sides $=10$, 6 sides $=4$ 5 sides $=9$, 5 sides $=3$ 6 sides $=8$, 4 sides $ …

It's not true that it works for $Z$ small enough. Consider the $2 \times 2$ case $$ Z = \pmatrix{t & 2t\cr 2t & 4t\cr} $$ which is positive semidefinite for $t \ge 0$. Then $$\det(X) = \log(1+t)\log( …

Here's a very simple example which illustrates one of the difficulties of this approach. Consider $t=1/2$ and random variables $X$ which is $1/2$ with probability $1$ and $Y_\epsilon$ which is $1/2 + …

$$\eqalign{G(s) &= \sum_{j=0}^\infty \dfrac{a^j (s - 1)^{2j}}{j!} = \sum_{j=0}^\infty \sum_{n=0}^{2j} {2 j \choose n} \dfrac{(-1)^{2j-n} a^j}{j!} s^n\cr & = \sum_{n=0}^\infty \sum_{j = \lceil n/2 \rc …

The equality constraints determine an affine subspace of $\mathbb R^n$. After suitable change of variables, this becomes $\mathbb R^m$ for suitable dimension $m$, and the nonnegativity of the entries …

If $X$ and $Y$ have joint density $f(x,y)$, we have $p(q,\alpha) = \int_{-\infty}^\infty dy\ \int_{-\infty}^{q - \alpha y} dx\ f(x,y)$ and thus (assuming sufficient regularity) $\partial_\alpha p(q,\a …

So your random variables $X$ and $Y$ have Gamma distributions with scale parameters $x_0$ and $y_0$ and shape parameters $1-\tau$ and $1-\kappa$ respectively (of course, you must assume $\tau < 1$ and …

Note that the indicator $$ 1_{X_i = x} = \frac{1+x X_i}{2}$$ so that $$ \mathbb P(X_i = x_i,\; X_j = x_j) = \mathbb E \left[ \frac{(1+x_i X_i)(1+x_j X_j)}{4} \right] $$ Expand it out, and note that b …

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 …