# Questions tagged [limit-theorems]

For questions about limit theorems of probability theory: (functional or not) central limit theorem, law of large numbers, law of iterated logarithm, etc.

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### Finding a sequence from weak convergence

Let $(X_n)_n$ be a sequence of independent random variable, $(u_n)_n$ a sequence of positive numbers, such that $$\frac{1}{u_n}\sum_{k=1}^nX_k \Rightarrow X$$ where $X$ is not degenerate. Prove that ...
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### Stable law and the domains of attraction

The multivariate generalised central limit theorem with their domains of attraction was given by Rvačeva (see also this post). The original paper is not very accessible on the internet, and neither ...
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### Interplay between CLT and convergence in Total Variation

Given a random variable $X$ with bounded moments such that $E[X] = 0, E[X^2] = 1$, let $F_n$ denote the distribution $\sum\limits_{i=1}^d\frac{X_i}{\sqrt{n}}$ where each $X_i$ is an independent copy ...
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### Strong law of large number for semimartingale

I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion. Thanks
I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws. Let $... 0answers 86 views ### Functions whose Laplace transforms have prescribed behavior at minus infinity Let$f:\mathbb{R}\rightarrow\mathbb{R}$be a non-negative function with entire Laplace transform$\hat{f}$(in particular$\lim_{t\to \infty}e^{st}f(t)=0$for all$s$), and$p_0$a positive integer. ... 0answers 319 views ### Unusual generalization of the law of large numbers I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ... 4answers 5k views ### Rate of convergence in the Law of Large Numbers I'm working on a problem where I need information on the size of$E_n=|S_n-n\mu|$, where$S_n=X_1+\ldots+X_n$is a sum of i.i.d. random variables and$\mu=\mathbb EX_1$. For this to make sense, the$(...
Let us say have a sequence of $n$ 2-$D$ random variables $X_i=(\varepsilon_i/\sqrt{n},i\varepsilon_{i}\sqrt{6}/n^{3/2})$, where $\varepsilon_{i}$ are independent random variables such that \$\mathbb{P}(...