# Tagged Questions

Convergence of series, sequences and functions and different modes of convergence.

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### Limits of an indeterminate form $\lim_{t\to\infty} (a+b(-m)^t)/(c+d(-m)^{t-1})$ [on hold]

I'm trying to solve the limit of the following indeterminate form: $$\lim_{t\to\infty} \frac{a+b(-m)^t}{c+d(-m)^{t-1}}$$ where $t=1, 2, 3, \cdots$ denotes time and all the coefficients are positive ...
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### markov processes and ergodic theory

For an ergodic Markov Chain $$\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]$$ where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
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### Heights of multiples of rational points on elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve, given by some minimal Weierstrass equation (say $Y^2 = X^3 + aX + b$ for some integer $a$ and $b$), and let $P$ be a rational point on $E$ which is not the ...
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### Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
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### Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
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### On a matrix algorithm involving rank-one projections

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration X_{k+1}=\frac{1}{N}\sum_{i=1}^...
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### Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram

Consider the direct limit of an indexed family $\{a_n\}_{n\in \omega}$: $\require{AMScd}$ \begin{CD} a_0 @>>> \ldots @>>> a_n @>>> a_{n+1} @>>> ...
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### When does this interesting sum diverge?

For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$ I don't know of any references or methods for this -- not even for $x=1$, for which the ...
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### Lemma in Scholze-Weinstein

In the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein (see http://math.bu.edu/people/jsweinst/Moduli/Moduli.pdf), one finds the following claim in Lemma 5.2.7: Lemma: Let $K$ be a ...
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### Is $\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$ finite for every $k$?

I would like to check if this limit :$$\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$$ finite for every $k$? where :$\phi_{k}$ is iterating Euler - totient function ...
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### 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 ...
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### Almost sure convergence of smallest eigenvector of diagonal matrix

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily ...
I encounter an optimization problem. The simplified version is like following: Denote function $F(x):\mathbf{R}^n\rightarrow\mathbf{R}$, where $F(x)$ is a smooth lower bounded convex function (i.e. $... 1answer 92 views ### Pointwise convergence of polynomials to a function on a compact set K that is 1 on some disc D and zero outside D Motivation of my question: Let$A$be a bounded selfadjoint operator with spectral measure$E$and$x$a vector. Then it is easily seen that the closed linear span of all$A^nx$($n\in\mathbb N$) ... 0answers 62 views ### Proving convergence is impossible for a sum of hyperbolic cosines Suppose that$z$is some complex value. Is it possible to prove that $$\lim_{n \rightarrow \infty} \sum_{j = 1}^n {\sqrt{n \over j}} \cdot \cosh(z \log {n \over j}-\operatorname{ Arccoth} (2z))$$ ... 1answer 104 views ### Literature question on the convergence rate of the empirical distribution Assume that given$n$i.i.d samples$(X_1, X_2, ..., X_n)$drawn from$p_X$, an unknown probability mass function defined over a finite alphabet$\mathcal{X}$, one wants to estimate$p_X(x)$for each$...
In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as:  \begin{...