# 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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1 vote
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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 ...
1 vote
153 views

### 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 $Z_1, Z_2, \dots, Z_n$ ...
1 vote
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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 ...
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 ...
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, ...