Questions tagged [limits-and-convergence]
Convergence of series, sequences and functions and different modes of convergence.
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How to prove the convergence of the maximum point random variable of random concave function sequence?
I am really wondering how to prove this lemma from the book 'Counting processes and survival analysis'. No need for the first and second point, just the third point, why does the maximum random ...
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Weak convergence of probability measures implies convergence of integrals of bounded functions that are continuous almost everywhere?
Let $(X, \mathcal{B})$ be a compact metric space, and let $(\mu_n)_{n\ge 1}$ be a sequence of probability measures on $X$ such that $\mu_n$ converge weakly towards $\mu$.
Let $f : X \to \mathbb{R}$ be ...
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QR code problem
Motivation. Today, I saw a QR code with an unusually large black square (a largish group of "pixels" coloured black and forming a square). This inspired the following problem.
Problem. Fix $n\in\...
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Uniform integrability for nets of measures under weak convergence
Let $X$ be a topological space (or metric space if needed) and let $(p_t)_{t\in T}$ be a net of Borel probability measures on $X$ which converges weakly to a Borel probability measure $p,ドル that is, ...
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1
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Limit of a sequence defined via return frequencies to a measurable set
Let $(X, \mathcal{B}, \mu)$ be an arbitrary measure space and $T: X \to X$ a measure-preserving transformation. Fix a measurable set $A \in \mathcal{B}$ with 0ドル < \mu(A) < \infty,ドル and fix $t \...
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70
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Convergence of Kaplan-Meier estimator for pooled sample
I am new to survival analysis. Recently I have been thinking about the Kaplan-Meier estimator for pooled sample. Suppose we have two group of samples, group 1 has $n_1$ samples from the survival ...
0
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1
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157
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Lattice sum estimates (potentially via integrals) - Probably diverging
If we have $f:\mathbb R^d\to\mathbb R_{>0}$ of the form
$$f(x):=\frac{1}{|a+i(\|x\|^2-\|x-K\|^2+b)|^p},$$
where $a\in\mathbb R\setminus\{0\}$ and $b\in\mathbb R$ are real numbers, $K\in\mathbb R^d\...
0
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1
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172
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Lattice sum estimates (potentially via integrals)
If we have $f:\mathbb R^d\to\mathbb R_+$ of the form
$$f(x):=\frac{1}{|a+i(\|x\|^2+\|x-K\|^2+b)|^p},$$
where $a\in\mathbb R^*$ and $b\in\mathbb R$ are real numbers, $K\in\mathbb R^d\setminus\{0\},ドル $i$...
5
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Weak convergence of measures in Hilbert space and convergence of norms
Let $H$ be a separable infinite-dimensional Hilbert space over $\mathbb{R}$ with norm $\|\cdot\|$.
Let $\{\mu_\alpha\}$ be a net of Borel probability measures on $H,ドル and let $\mu$ be a Borel ...
1
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1
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129
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How to prove weak convergence of a truncated Brownian motion process on D[0,$\infty$] with uniform topology
I am going through some basic stochastic process weak convergence theory and really need your help. Thanks! The question might be very naive. Suppose $B(t)$ is the standard Brownian Motion. If $f(x)$ ...
2
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1
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$ \lim_{n \to \infty} \left(1 + \sum_{i=2}^{n} \left( \sum_{k=i^2-(i-1)}^{i^2} \frac{1}{k} \right) -\ln(n)\right)$
I'm exploring a sequence defined by a specific sum of reciprocals of integers. Let $a_n$ be this sequence, where each term is defined as:
$$a_1=1$$
$$a_n = 1 + \sum_{i=2}^{n} \left( \sum_{k=i^2-(i-1)}^...
5
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1
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Uniform convergence of metrics imply that induced topologies are equal and possible error in Burago, Burago, Ivanov's "A Course in Metric Geometry"
Suppose $X$ is a set and $d_n,d$ metrics on $X$ for $n \in \mathbb{N}$ and let $\tau_{d_n},\tau$ be the induced topologies on $X$. Suppose that $d_n \to d$ uniformly as $n \to \infty$.
Does this imply ...
2
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0
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166
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Probability mass control of branching processes?
I am interested in any knowledge on this question:
The process starts with a single particle at time 0ドル,ドル then after a random time of distribution $G$ which has a density on $(0, \infty)$ it gives ...
14
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1
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Summation convergence that has numerical evidence but has not been proven analytically
I have two expressions. The first one \eqref{1} is a summation: as the quantity $M$ approaches infinity, it converges to the second \eqref{2} expression. This behavior has numerical evidence: ...
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172
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In which cases Hahn-Banach theorem holds for pseudotopological Hausdorff locally convex linear spaces?
In which cases Hahn-Banach theorem holds for pseudotopological Hausdorff locally convex linear spaces? I would be grateful for references.
Some definitions for context.
Pseudotopological space is a ...