# Questions tagged [limits-and-convergence]

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

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### LLN of random nearest neighbor function

There are two samples of iid random variates: $X=\{X_1,X_2,...,X_n\}$ and $Y=\{Y_1,Y_2,...,Y_n\}$. Further, $\forall i,j: X_i$ is independent of $Y_j$. The probability distributions $P,Q$ are unknown ...
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### Convergence in probability of series of random variables

From the standard literature it is well known that for sequences of random variables $X_{1, n} \stackrel{P}{\rightarrow} X_1$ and $X_{2, n} \stackrel{P}{\rightarrow} X_2$ as $n \rightarrow \infty$ it ...
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### Convergence of localic maps

We can define a limit of a sequence of points in a locale in the usual way: $x$ is a limit of $\{ x_i \}_{i \in \mathbb{N}}$ if, for every open $U$ containing $x$, there exists $N$ such that $x_n$ ...
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### $|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)|=O_P(\frac{1}{\sqrt{n}})$ under $E(|X_1|)<\infty$?

For i.i.d. random variables $X_1,\dots, X_n$ with $E(|X_1|)<\infty$. Does the following equation hold? $$\left|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)\right|=O_P\left(\frac{1}{\sqrt{n}}\right)$$ I ...
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### Limiting behavior of a sequence of polynomials

Let $f(z)\in\mathbb{C}[z]$ have all its zeros on the line $\Re(z)=\alpha$ for some $\alpha\in\mathbb{R}$. It is an elementary fact (equivalent to Lemma 9.13 here) that if $u\in\mathbb{C}$ and $|u|=1$, ...
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### Convergence radius of double series with Pochhammer symbols

I would like to know the convergence radius of the following two double power series of $(x,y) \in \mathbb{C}^2$: \begin{align} \sum_{m,n=0}^\infty \frac{(d-a)_{n+m}(d+b)_{n+m}(d+a)_n(d-b)_n}{n!m!(2d-...
### Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?
The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions : Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...