# Questions tagged [limits-and-convergence]

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

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### Find limit of sequence defined by sum of previous terms and harmonics

I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function? I tried to ...
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### Convergence acceleration of a series by using optimal parameters

One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...
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### Comparison of several topologies for probability measures

Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
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### Linear programming with a convergent coefficient

The following linear programming problem $x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$ has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...
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### Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes

Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
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### Asymptotic rate for the expected value of the square root of sample average

I have iid random variables $X_1, \dots, X_n$ with $X_i \geq 0$, $E[X_i]=1$ and $V[X_i] = \sigma^2$. Let $S_n = \frac{\sum_{i=1}^n X_i}{n}$. I'd like to say that $E[\sqrt{S_n}] = 1-O(1/n)$. My first ...
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### Integrals of power towers

Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
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### If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^*$ = $\mathfrak{M}(G)$

Let $\mathfrak{M}(G)$ be the set of all means on $L_\infty (G)$ If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^* = \mathfrak{M}(G)$ My attempt: We know ...
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### Does this definition of the Fourier intensity measure make sense?

Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$. For simplicity, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$ (although the following definition is ...
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### How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?

Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...
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### Showing $o(1)$ convergence for ratio of successive binomial tail probabilities

For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that $$\frac{P(X>c)}{P(X>c-1)}=1-o(1)$$ uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...
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### Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
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### Asymptotic distribution of $n\mathbb E_{\hat{P}_n}[g(Z;\theta)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta)]$

Setup This question is a followup on this question. I'm interested in the asymptotic distribution of certain quadratic forms. So, let $Z$ be a $p$-dimensional random vector with (unknown) ...
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### Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
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### A convergence condition on tempered representation

Assume $\pi$ is a tempered representation of $GL_n(\mathbb{Q}_p)$. $N_n$ is a maximal unipotent subgroup of $GL_n$, and $\xi$ is a non-degenerate character of $N_n(\mathbb{Q}_p)$. Let $\Pi$ be the ...
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### Positive splitting of Sobolev convergence

Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions wuch that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
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### Convergence of SDEs

Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
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### Convergence Based on Recurrence Relation

I am studying a sequence based on the following recurrence: $$X[t] = \sqrt{\alpha X[t-1]^2+(X[t-1]^2-\alpha X[t-2]^2)\frac{(2-X[t-1])^2}{X[t-1]^2}}$$ $$X=0$$ $$X>0$$ $$\alpha \in (0,1)$$ I ...
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$0<\lim_{\beta\rightarrow 1}(1-\... 0answers 60 views ### Limits of a simple damped system Definition: Let F_n(s) = \frac{1}{s^{n+1}(1+s)^n} be the Laplace transform of f_n(t). Required Result: To show \lim_{n\rightarrow\infty}f_n(n+n/e) < o(n). Ideas: Let G_n(s)=\frac{1}{s^{n+... 1answer 209 views ### Intuition for coercive functions I have been working with \Gamma-convergence for some time now; it has lead me to wonder: What is the intuition behind coercive functions? 3answers 176 views ### Asymptotic forms of Legendre functions for large degree Does anyone know where to find (or how to obtain) expressions for the Legendre functions for large degree, to second order? For example, to first order the expressions are$$ P_n(\cosh(x)) ~ \...
Let us suppose that $X_1,\ldots,X_n$ with $n\ge1$ are iid random variables such that $\operatorname EX_1=0$ and $\operatorname E|X_1|^s<\infty$ with some $s>2$ and define the DFT of \$X_1,\ldots,...