We study the simultaneous zeros of a random family of $d$ polynomials in $d$ variables over the $p$-adic numbers. For a family of natural models, we obtain an explicit constant for the expected number of zeros that lie in the…

The improper stochastic integral $Z= \int_0^{\infty-}\exp(-X_{s-})dY_s$ is studied, where ${ (X_t ,Y_t) , t \geq 0 }$ is a L'evy process on $R ^{1+d}$ with ${X_t }$ and ${Y_t }$ being $R$-valued and $R ^d$-valued, respectively. The condition for existence…

Chung, Diaconis, and Graham considered random processes of the form $X_{n+1}=2X_n+b_n \pmod p$ where $X_0=0$, $p$ is odd, and $b_n$ for $n=0,1,2,\dots$ are i.i.d. random variables on $\{-1,0,1\}$. If $\Pr(b_n=-1)=\Pr(b_n=1)=\beta$ and $\Pr(b_n=0)=1-2\beta$, they asked which value of $\beta$ makes $X_n$…

Let $F$ be a probability distribution with support on the non-negative integers. A model is proposed for generating stationary simple graphs on $Z$ with degree distribution $F$ and it is shown for this model that the expected total length of…