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My colleague Ed Perkins used quite a bit of nonstandard analysis in probability theory in the early 80's. See for example http://www.springerlink.com/content/e636h42166202387/ I don't know if he's us …

Perhaps the main "analytic" area in Quantum Field Theory is known as Constructive Quantum Field Theory. This essentially emerged in the 1960's with the Wightman axioms. There is still work going on t …

Although in principle the sample space, with its $\sigma$-algebra and probability measure, comes first, things are not always so neat in real life. In applications it is often the random variables ( …

$\text{tr}(AB+BA) = 2 \operatorname{tr}(A^{1/2} B A^{1/2}) > 0$, so that may produce some bias toward positive eigenvalues. In particular if you generate your "random" matrices in such a way that the …

More generally, you could ask this for any irreducible Markov chain and any starting state. For each nonempty set S of vertices not containing the starting state $s_0$, let $T_S$ be the time (in steps …

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 …

Let $f(x) = \sum_{j=0}^d a_j x^j$ be any polynomial of degree $d$ such that $f(0) = 0$ and $f(x) \le 1$ for all $x \ge 0$. Then $$P(X > 0) \ge E[f(X)] = \sum_{j=0}^d a_j E[X^j]$$ Your Cauchy-Schwarz …

Since you can scale everything linearly, let's suppose $\mathbb E(X) = \mathbb E(Y) = 1$. I'm not sure this is optimal, but I think it must be close. Consider a case where all but two of the points a …

Let $R(r,p,s)$, $P(r,p,s)$, $S(r,p,s)$ be the probabilities of rock, paper, scissors with initial distribution $r,p,s$, where $r + p + s$ is a power of $2$. Now for $r+p+s = 2^{n+1}$, rock will win t …

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) = …

If we can find independent random variables $U_j$ uniform on $[0,1]$ we can transform them to $\mathcal N(0,1)$. If $Y$ is uniform on $[0,1]$, let $D_j$ be its $j$'th decimal place, i.e. $Y = \sum_{j …

Your matrix $A = X^T X$ where $X$ is a random $m \times N$ matrix with a continuous distribution having a density. An $m \times m$ submatrix of $A$ is $Q^T A P = (XQ)^T XP$ where $P$ and $Q$ are $N \ …

For $k=3$ with $N$ not divisible by $3$, the probability of a tie is $$ \sum_{m= \lceil N/3 \rceil}^{\lfloor N/2 \rfloor} \dfrac{N!}{m! m! (N-2m)!} 3^{1-N}$$ The sum seems not to have a closed form. …

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 - …

1) No, it won't. Suppose $\mu_j(t)$ is the probability vector at time $t$. Let explosions happen at rate $r(t)$ at time $t$. Then we should have $$ \dfrac{d}{dt} \mu_j(t) = -3^j \mu_j(t) + (2/3) 3^ …