# Questions tagged [probability-limit-theorems]

For question about limit theorems of probability theory, like the law of large numbers, central limit theorem or the law of iterated logarithm.

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### Interchanging infinite sum and limit in distribution

I'm trying to do a proof for a project and I've run into the following problem. For each $j$ consider a sequence $(Y_{j,n})_{n \in \mathbb{N}}$ of random variables such that the different sequences ...
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### Limit for Brownian local time

If we denote $(B(t))_{t\geq 0}$ as the standard Brownian motion, as by the scaling property, we know for every $n\in\mathbb{N}$, we have $\frac{B(nt)}{\sqrt{n}}\overset{(d)}{=}B(t)$, and denote for ...
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### The function $\log^+x=\max\{1, \log x\}$.

I was reading Marcinkiewicz-Zygmund (MZ) law of large numbers for random fields and came across necessary and sufficient condition $E(|X|\log^+|X|)< \infty$ for MZ-SSLN to hold true. I have a ...
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### Infinite divisibility and Gaussian random variables

I was looking for a simple explanation of why the Gaussian random variable can be the only distribution appearing in the Central limit theorem. From the statement of the Central limit theorem, it is ...
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### Convergence of positive random vector

Suppose I have a sequence of positive random vectors $\vec{X}_N$ of fixed length $l$. That is, $\vec{X}_N = (x_N^{(1)}, x_N^{(2)},\cdots, x_N^{(l)})$ where each entry $x_N^{(i)} > 0$. Suppose I ...
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### Proving convergence in prob of $X_n = X = Y_n$ using Markov's inequality.

Below question is from the book 'Probability course.com'. The book provides a solution using Chebyshev's inequality. Before reading that solution, I used Markov's inequality. Is my solution correct? ...
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