# Questions tagged [limits-and-convergence]

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

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### Continuity of an upper semi-continuous function over periodic points

Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...

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### Show that $\int_{\mathbb R^p} x \nu(dx)=0$ in a question related to a certain convergence of measures

Consider the sequence of estochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{I}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...

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### Characterizing functions that are limits of integrable lower-bounded functions

Let $X$ be a separable Hausdorff topological space, endowed with a positive finite Borel regular measure. Consider those (trivially measurable) functions $f : X \to \mathbb R$ such that their ...

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### Properties of statistical estimators when data is a collection of estimates

Assume I have a statistical estimator $\theta$ that has nice properties (say, unbiased and consistent) when the data $Y=\{y_1,y_2,\dots,y_n\}$ is i.i.d. (possibly with additional assumptions). But now,...

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### Error bound for stohastic gradient descent method

To solve an optimization problem $\min_x G(x)$ using standard stochastic gradient descent method, we let $x_0$ be the initial point and $x_k$ be the $k$-th point such that
\begin{equation}
x_k = x_{k-...

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### How "correct" is Knuth's fast addition $(a,b) \mapsto (a \oplus b) \oplus ((a\land b) \ll 1)$?

Donald Knuth suggested a bitwise approximation for addition on the non-negative integers that is very fast on common processors:
$(a,b)\mapsto (a\oplus b) \oplus ((a\land b) \ll 1)$,
where $a,b$ are ...

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### Sequence of $L^2$ functions converging to zero weakly s.t. $|f_n|^2$ converges to 1 weak-star?

I am trying to construct a sequence $\{f_n\} \in L^2([0,1])$ with $f_n \geq 0$ a.e. such that $f_n \to 0$ weakly in $L^2$ (meaning $\int f_n dx \to 0$ for all $f \in L^2([0,1]))$ and such that for all ...

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### Asymptotic scaling of mean and variance for the norm of random vector (non gaussian components)

The norm of a vector whoose components $X_i$ are normally distributed follows the Non-central chi distribution and it can be shown that, increasing the number of components $k$ (i.e. the dimension of ...

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### Meromorphic extension of a limit function

Suppose $f_j(z)$, $j=1,2,..$ is a sequence of meromorphic functions on the complex plane $\mathbb{C}$. With a common set of all poles given by $S = \{-i,-2i, -3i,..\}$.
Assume that each of them is ...

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### Can we modify this extended pseudometric such that its convergence is equivalent to that in measure?

Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...

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### Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$

Let $f = (f_0,f_1,\ldots,f_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum_{n\geq 0} z^n f_n$$ for its probability generating ...

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### Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers

This is a repost from MSE because I got no answers there.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...

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### Convergence of random variables based on shifts of a markov chain

Suppose we have a discrete time (not necessarily stationary) Markov chain $X=(X_0,X_1,X_2,\dots)$ on $(\Omega, F)$. We assume $X$ is Harris ergodic with an invariant distribution.
Suppose we have a ...

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### Convergence of the perimeter of level sets

I have already posted this question on Math StackExchange. Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}...

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### Quantitative version of ergodic theorem in Markov chains

Consider an irreducible Markov chain $X_t$ with finite state space $E$, and unique invariant measure $\pi$. Fix a function $V:E\to\mathbb R$ such that $E_\pi[V]=0$. The ergodic theorem tells us that, ...

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### Convergence of the row sums in a triangular null array with zero mean

Let $(X_{jn})_{1\leq j \leq n}$, $n\in \mathbb N$, be a triangular array of random vectors in $\mathbb R^d$ (the $X_{jn}$ are understood to be independent in $j$ for fixed $n$.). We say that the ...

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### Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?

This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...

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### Does $\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert \, dt$ converge for $L^1_\text{loc}$ $f : [0,\infty) \to \mathbb{R}$?

Let $f(t) : [0,\infty) \to \mathbb{R}$ be an $L^1_\text{loc}$ function.
Then, I wonder if the following series
\begin{equation}
\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert
\, dt
...

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2
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### Reference request and clarification for Central Limit Theorem for complex random variables

I'm looking for a reference and a proof of the following version (or eventually a more general version) of the Central Limit Theorem for complex random variables.
Theorem. Let $Z_1, Z_2, \dots, Z_n$ ...

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### Central limit theorem with limit of functions

Suppose that
$$\sqrt{n}(X_n - \theta)\xrightarrow{d} X,$$
according to the delta method, we have
$$\sqrt{n}(g(X_n)-g(X))\xrightarrow{d} g'(\theta)X$$
when $g$ is differentiable.
My question is, if
$$\...

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### $\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}$ for various $x$

Let $$f(x)=\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}.$$
Compute $f(1)$ and $f(2)$.

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### Convergence of stochastic linear recurrences

Suppose that $\zeta_t$ is a univariate, stationary stochastic process ($t\in\mathbb{N}^+$).
Let $x_0\in\mathbb{R}^n$, and let $f:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a continuously ...

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1
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### A question about convergence of stochastic processes converging to a random walk

Consider the following random walk $(y_t)_{t \in \mathbb Z_+}$:
$$y_t = y_{t-1} + u_t,\quad (u_t)_{t \in \mathbb Z_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z_+)$$
where $y_0, u_1, u_2,...$ ...

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1
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### Convergence rate of a sequence of sets to a set-theoretic limit?

Suppose $n\in\mathbb{N}$ and set $A\subseteq\mathbb{R}^{n}$.
If we define a sequence of sets $\left(F_r\right)_{r\in\mathbb{N}}$ with a set theoretic limit of $A$; how do we define the rate at which $\...

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### Numerical strategies for evaluating a modular invariant infinite sum

I'm working on a problem that involves the numerical evaluation of the following infinite sum:
$$
\sum_{m=-\infty}^{\infty} \ln \left|1\pm e^{-2\pi \tau_1 \sqrt{m^2+x^2/(4\pi^2\tau_1)}-2 \pi i \tau_0 ...

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1
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### calculating a double limit

We have the following term:
$$ (e^{-a h}+e^{-b h})^n / 2^n$$
Now we take the limit:
$$ h\to 0, n\to \infty $$
What relation of $h$ and $n$ must be satisfied for the following limit to hold?
$$\lim_{h\...

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### Rate of convergence of Fejer kernel to the Dirac delta function

This seems like something one might find in a book so I would be grateful for any references you think may be helpful.
I am interested in the rate at which of a function integrated against the $N$th ...

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### Convergence in expectation of a discontinuous function

Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...

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### Do finite exchangeable random sequences behave asymptotically independently?

Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ and $(Y_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be triangular arrays of row-wise exchangeable random variables, that is for any $n$ and permutation $\...

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### On the spectrum of Fokker–Planck with linear drift

The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...

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### convergence for series of random variables

Suppose $X_n\sim N(0,1) $ is iid, then it is easy to see that
$$\sum_{n=1}^{\infty}\frac{X_n}{n}\cos nx$$
converges a.s. for any $x$ since
$$\sum_{n=1}^{\infty}var(\frac{X_n}{n}\cos nx)<\infty$$
...

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### Surprisingly difficult limit of a sequence

Is there an easy way to prove that $|\operatorname{Re}(a_n)| \to \infty$ where $a_n=\left(\frac{1}{2}+i\frac{\sqrt{7}}{2}\right)^n$?
Of course $|a_n| \to \infty$, but we have
$$
\operatorname{Re}(a_n)=...

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### On Zagier's missing continued fraction with multiple limits?

I. Zagier's continued fraction
As pointed out by Gorodetsky in his answer, Zagier evaluated the continued fractions associated with his six sporadic sequences excepting the one for $(-9,-3,-27)$. Let $...

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### Finding a new level-$7$ pi formula using the relation $j_{7A}(\tau) = \left(\sqrt{j_{7B}(\tau)}+\frac{7}{\sqrt{j_{7B}(\tau)}}\right)^2$

(Note: This third method continues from this post.)
There are level-$7$ pi formulas based on the McKay-Thompson series $T_{7A}$ and Cooper's $s_7$ sequence in this paper. This third method, among ...

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### Ramanujan's pi formulas with a twist (nine years later)

(Note: The second method described here continues this post.)
About nine years ago, I made an MO post "Ramanujan's pi formulas with a twist". An answer was informative, but not completely ...

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### Transformations of Ramanujan's 1/pi formulas $\sum_{n=0}^{\infty} s(n)\frac{An+ B}{C^n}$ and Monster moonshine functions

Someone with many papers on Ramanujan's work asked me how I managed to find the closed-forms for the binomial sums of level $10$ in a recent MO post. (A colleague of his wasn't able to find them.) I ...

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### Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped

I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...

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### Finding a distance so that this function is a contraction mapping

Let $f(x,y)=(y,\frac{2}{x+y})$ defined on $(0,\infty)\times (0,\infty)$. Is there a distance $d$ on $(0,\infty)\times (0,\infty)$ such that $f$ is a contraction of the metric space $((0,\infty)\times (...

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### Convergence of Solutions of Integral Equations with Weakly Converging Forcing Terms

Let $\Omega$ be a bounded interval of $\mathbb{R}$ and let $y\in L^\infty(\Omega \times (0,T))$ be a mild solution of the integral equation
$$
y(\cdot,t)=S(t) y_0+\int_0^t S(t-s) \left[u(\cdot,s)y(...

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### Which kind of convergence can we get from Laplace transform convergence?

This question is a related question see this post Vague convergence VS Laplace transform convergence. But now we assume that
\begin{equation}
\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}...

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### Vague convergence VS Laplace transform convergence?

If we assume that $\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu_n\to\mu$ vaguely. Where $\mu_n$ is a measure. Please check here for ...

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### Bound the expectation of an average

Let $(a_n)_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu_l(b) \le P[a_n = b\mid a_1,\ldots,a_{n-1}] \le \nu_u(b)$ almost surely for every $n \ge 1$ and $b \in ...

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### Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$

When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...

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### Fischer Information and Entropy, matrix case, determinant of covariance matrix going to zero

If $X \sim \mathcal{N}(\mu, \sigma^2)$, then
\begin{equation}
\mathcal{I}\left(\mu, \sigma^2\right)=\left(\begin{array}{cc}
\frac{1}{2\mathcal{H}...

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### Computing a limit on the unit sphere: Riemann Lebesgue?

Let $u\in L^1(\mathbb{S}^{d-1})$. I want to show that
\begin{align*}
\lim_{|\xi|\to \infty}
\int_{\mathbb{S}^{d-1}}(1-\cos(\xi\cdot w))u(w)d \sigma_{d-1}(w)
= \int_{\mathbb{S}^{d-1}}u(w)d \sigma_{d-1}(...

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1
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### Finding weak LUR property of $C[0,1]$ with an equivalent norm

On the space $X=C[0,1]$, define a norm $||| f |||^2=\Vert f \Vert_{\infty}^2 + \Vert f \Vert_2^2$, where $\Vert \cdot \Vert_\infty$ is the sup norm on $C[0,1]$ space and $\Vert \cdot \Vert_2$ is the $...

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### proving inequality in Riemann zeta function

Recently I have made some interesting observations on the limit $$\lim_{k\rightarrow \infty}{\sum_{n=1}^{k}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}}. $$ When this ...

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### $\operatorname{Coth}(\alpha_n a) \to i$ when $n \to \infty $

I have a question about the following statement from the article
Alves, M., Rivera, J.M., Sepúlveda, M., Villagrán, O.V. and Garay, M.Z., The asymptotic behavior of the linear transmission problem in ...

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1
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### How to prove approximation for fresnel integral converges

I was looking at the fresnel integral $S(x)=\int^x_0\sin(s^2)ds$. From reading I learned that this integral approaches $\frac{1}{2} \sqrt{\frac{\pi}{2}}$ as $x \rightarrow \infty$. Through messing ...

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3
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### Solving a limit about sum of series

what's the limit of
$\sqrt{1-t}\sum _{n=0}^{\infty}t^{n^2}$ as $t$ goes to the left of $1$? i.e. $t\to 1^{-}$? I tried several times but failed. Here is my thought:
This is a $0\cdot\infty$ problem, ...