## Convergence of a particular double sum [closed]

Closed. This question needs details or clarity. It is not currently accepting answers. Want to improve this question? Add details and clarify the problem by editing this post. Closed 6 years ago. Improve this question Consider the following double sum: Q(n)=1n2n∑i=1n∑j=1[∂ijlnf(x)]2 where ∂ij is the partial second order derivative (bounded for all indices), the function … Read more

## On contractive properties of a nonlinear matrix algorithm

I’m stuck in a problem that concerns a nonlinear iterative matrix algorithm. Although the problem is quite complicated to explain I’ll try to describe it in a simple way, neglecting unnecessary details. The matrix iteration is the following one: Xk+1=∫π−πX1/2kG(ejω)Ψ(ejω)G⊤(e−jω)XkG(ejω)G⊤(e−jω)X1/2kdω2π(⋆) where {Xk} is a sequence of n×n matrices, and G(ejω), Ψ(ejω)=Ψ⊤(e−jω) are n×1 and 1×1, … Read more

## Interplay between CLT and convergence in Total Variation

Given a random variable $X$ with bounded moments such that $E[X] = 0, E[X^2] = 1$, let $F_n$ denote the distribution $\sum\limits_{i=1}^d\frac{X_i}{\sqrt{n}}$ where each $X_i$ is an independent copy of $X$. It is known that as $n \to \infty$, $F_n$ converges in distribution to a standard Gaussian. The Berry-Esseen inequality then gives us a quantitative … Read more

## Limits of definable maps

For sequences of semialgebraic maps there is the following result: Let (fn:]0,1[d→]0,1[)n∈N be a sequence of continuous semialgebraic maps of bounded degree such that (fn)n∈N converges uniformly to some map f. Then f is a continuous semialgebraic map. I wonder (and doubt) if a similiar statement is true in o-minimal expansions of the reals ¯R=(R,<,+,∗,0,1), … Read more

## Convergence in Product Formula for Tamagawa Number

Let G=SLn, and recall that its Tamagawa number is τ(G)=1 and is given by the product expansion τ(G)=Vol(G(Z)∖G(R))⋅∏pVol(G(Zp)), where for each prime p we have that Vol(G(Zp))=|G(Fp)|pn2−1=pp−1⋅n−1∏i=0(1−pi−n)=n−2∏i=0(1−pi−n). I believe can show that Vol(G(Z)∖G(R)) is bounded independent of n. Indeed, one can prove that n−2∏i=0(1−pi−n)−1≤1+p−3/2 by comparing the Maclaurin series expansions of the logarithms of both … Read more