8 problems

Let $k, ell\ge 0$ and $\beta,z>0$. Define … and let $r_{(k,\ell),\beta}(z)$ be the corresponding limiting integral ratio. Second-term oscillation conjecture. As $n\to\infty$, the q…

Let $q\geq 2$, and let $p_0,\ldots,p_{q-1}$ be non-negative numbers adding to $1$, with no $p_j=1$. Let $x_1,x_2,\ldots$ be independent, identically distributed random variables sa…

Let $Xa= sum{i=1}^n aiXi$ and $Xb=sum{i=1}^n biXi$, where $X1,ldots,Xn$ are i.i.d. Gamma random variables with parameter $alpha$, and let $aprec b$ denote majorization of the coeff…

Let $\alpha$ satisfy $-1<\alpha<-1/2$ in a generalized multicomponent Cuddeford system, and let $h(Q)$ be the function entering its distribution function. In the covered cases, the…

Consider a spherical stellar system with a consistent, positive phase-space distribution function. Let \gamma(r)=-d\text{{rm ln}, }\rho/d\text{{rm ln}, } r denote the density…

A spherical stellar system has density slope \gamma(r)=-d\text{{rm ln}, }\rho/d\text{{rm ln}, } r, anisotropy parameter $\beta(r)=1-\sigma_t^2/(2\sigma_r^2)$, and a positive…

Let $F_{l}^{{\zeta^*}}(x)$ be the limiting continuous distribution function associated with the parameter $l$. The source considers the separate positive and negative first moments…

Let $F^{{\zeta^*}}_{l,m}(x)$ be the distribution functions associated with the parameter $l$ and index $m$, and let $F_{l}^{{\zeta^*}}(x)$ denote the proposed limiting distribution…