# Questions tagged [limits-and-convergence]

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

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### Convergence in $\mathbb{L}_1$ implies convergence "perturbed" conditional expectations

Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following ...
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### Convergence in $H^{-2}$ of $L^2$-functions with limit in $L^2$

Assume a sequence $f_n$ in $L^2(\mathbb{R}^d)$ converges in $H^{-2}$ (w.r.t. its norm topology) to a limit $f \in L^2(\mathbb{R}^d)$. In this case, can one improve the convergence, for instance to ...
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### Is it known that the the sequence 7n+1 diverges to infinity starting with 7?

In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...
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### Convergence in probability of sample covariance for permutation invariant triangular arrays

Take two triangular arrays $X_{N,i}$ and $Y_{N,i}$ of random variables where $1 \le i \le N$. Suppose that the families $\{X_{N,i}\}$ and $\{Y_{N,i}\}$ are independent, and that the following ...
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### Uniform distribution as argument for copula likelihood

I am reading a well-known paper about copulas by Chen and Fan (2006). Specifically, Proposition 4.2 (see attached), in which all the arguments are uniform $U_{t-1}, U_t$. However, when the copula is ...
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### Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions

I asked this question on MSE here. Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
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### Power series expansions and limits of knot invariants

This question is moved from math stackexchange which I posted several days ago without an answer. Background(ignore this paragraph if you know finite type invariants well): Recall that a finite type ...
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### Does $\sum_{k = 0}^{qN} [e^{-\frac{\pi^2 k^2}{N}} - (\cos \frac{\pi k}{2 q N})^{8 q^2 N} ] \rightarrow 0$ as $q\rightarrow \infty$?

Let $N, q \in {\mathbb N}$. Let $S(N,q) = \sum_{k = 0}^{qN} [e^{-\frac{\pi^2 k^2}{N}} - (\cos \frac{\pi k}{2 q N})^{8 q^2 N} ]$. $N$ is fixed. Does $S(q,N)$ tend to zero as $q$ tends to infinity? How ...
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### Rate of convergence of the minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
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### Decay rate of minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
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### Uniform convergence of differential quotients in $L^1$

I know that the question arose already in other contexts. However, I think this question might be different. If I have $f\in W^{1,1}$ then it is obvious that $\frac{f(x+t)-f(x)}{t}$ converges ...
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### Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
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Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let \begin{equation*} E:=\bigcap_{n\ge0}B_n, \end{equation*} where \begin{equation*} B_n:=\...