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On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions derives the lower bound (theorem 1) $$\mathbb{P}\{ \text{det}(A_n) = 0\} \geq 2^{2-n} \binom{n …

I presume you want the coefficients $$a_k=\int_{-\infty}^\infty p(M)\psi_k(M)\,dM,$$ with $$\psi_k(x)=\frac{1}{\pi^{1/4}\sqrt{ 2^k k!}} e^{-x^2/2} H_k(x)$$ the normalized Hermite function of order $k$ …

Note that the random vector $u=(u_1,u_2,\ldots u_n)$, uniformly distributed on the unit sphere, can be replaced by the ratio $u=y/|y|$, with $y=(y_1,y_2,\ldots y_n)\sim N(0,I_n)$ a multivariate normal …

General remark on why one would study simple models: In the statistical mechanics of phase transitions one distinguishes relevant and irrelevant variables. A phase transition is associated with a dive …

Multi-dimensional generalizations (one player against $d$ other players) are explored by P. Lorek in Generalized Gambler's Ruin Problem: explicit formulas via Siegmund duality. For the analogue on a r …

Denote $p=mn$, then the mean $\bar{k}$ you seek equals $\bar{k}=\frac{p!}{(p-j)!}f_j(p)$, with $$f_j(p)=\sum _{k=1}^{\infty } \frac{k \mathcal{S}_k^{(j)} }{p^{k}}=\frac{F_j(p)}{\prod_{n=1}^j(p-n)^2}. …

A proof is on page 80-81 and 90 of Forrester, the probability distribution function of $W=X^\top X$ is $\propto e^{-\tfrac{1}{2}\operatorname{tr}W}(\operatorname{det}W)^{(n-m-1)/2}$, for an $n\times m …

Use the Euler angle parameterisation of the rotation matrix, $$Z=\begin{pmatrix} R(\alpha)&0\\ 0&1\end{pmatrix} \begin{pmatrix} 1&0\\ 0&R(\theta)\end{pmatrix} \begin{pmatrix} R(\alpha')&0\\ 0&1\end{pm …

The Hilbert-Polya approach to the Riemann hypothesis follows this path, by attempting to relate the zeroes of the Riemann zeta function to a quantum mechanical scattering problem. The probability dist …

Some pointers to the (extensive) literature on generalized Feyman-Kac formulas: Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian Fractional …

You want to derive the Fokker-Planck equation (the drift-diffusion equation for the density) from the Langevin equation (the stochastic differential equation for the position of a particle); this is s …

For elliptic PDE applications to options these would need be independent of time, they need to be perpetual (i.e. never expire), which is not a typical scenario. If your definition of "mathematical fi …

Since the Gaussian distributions of $w$ and $z$ are narrowly peaked around $x/2$, you can expand the logistic function around that value; to second order in the variance $\sigma_W^2+\sigma_Z^2$ this g …

Here it is, not the best quality scan, but it should serve the purpose. Urbanik and Woyczynski (1967) (the URL is also archived on the Wayback Machine, so it should last) I notice that some older pd …

This follows from the fact that $\mathbb{E}[A^\dagger A]=d I$ (with $A^\dagger$ the conjugate transpose of $A$ and $I$ the $d\times d$ identity matrix). Hence $$\mathbb{E}[\|C_n\|_F^2]=\operatorname{t …