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

Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?

Is there any closed form expression for the following serie? $$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$ Or at least a proof that it is an irrational number. The context of this problem is given by the following link: https://math.stackexchange.com/questions/2270730/whats-the-limit-of-sqrt2-sqrt2-sqrt2-sqrt2-sqrt2-sqrt2 In which it is proposed the problem of finding a closed form for the following nested radical: $$R = … Read more

Does rate of convergence in probability come from a metric?

In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is from the limit point. This can be done in a metric space as follows: Definition 1 (Rate … Read more

Set of subsequences with the same ultrafilter limit of the original sequence

Let U be a free ultrafilter on the positive integers N and fix U∈U such that U is not cofinite (thanks J.D.Hamkins for the correction.) Consider the natural bijection between (0,1] and the infinite {0,1}-sequences with infinitely many ones (written in base 2). Define also the sequence x=(xn) by xn=0 if n∈U and xn=1 otherwise, … Read more

Convergence of an=(1−12)(12−13)…(1n−1n+1)a_n=(1-\frac12)^{(\frac12-\frac13)^{…^{(\frac{1}{n}-\frac{1}{n+1})}}} [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 3 years ago. Improve this question I’m interesting to see the opinion of MO about my question which I posted here in SE, Answers I received have not convinced me, … Read more

Convergence acceleration of successions with logarithms

I have a numerical question regarding acceleration of a succession. A preliminary: suppose that I have a succession $a_g$ that, for high $g$, asymptotically goes as $$ a_g=s_0+\frac{s_1}g+\frac{s_2}{g^2}+…=\sum_{k=0}^\infty \frac{s_k}{g^k}. $$ I am interested in computing the coefficients $s_k$, but in particular I am interested in computing the leading coefficient $s_0$ (that would also be the … 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