# 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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### Question about random variables convergence

Further going through old lecture notes I've stumbled upon this... Let's say we are dealing with a sequence of random variables $\{X_n\}_{n=1}^\infty$ such that $\sqrt{n}(X_n-1)\to N(0,2)$ in ...
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### Confidence Interval based on limiting distribution of the Maximum Likelihood Estimator Sigma and Sample Variance

Another statistics question I need some help with. Now, I am not too familiar with limiting distributions (had my course on it over a year ago). I have tried some things myself and ran some ...
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### Conditions for a.s convergence of a gamma series

Let $(X_n)$ be a sequence of independent random variable, $X_n$ having the gamma distribution, $$f_{X_n}(x)=\frac{\alpha_n^{p_n}}{\Gamma(p_n)}e^{-x\alpha_n}x^{p_n-1}1_{[0,+\infty[}(x).$$ Find ...
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### Bounded probability implies convergence in probability

Let $(X_n)$ be a sequence of random variables and $(a_n),(b_n)$ be two sequences of non-negative real numbers such that $a_n\downarrow 0$ and $b_n\downarrow 0$ when $n\to\infty$. If for any $t>0$,...
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### An application of the convergence of types theorem

Let $(X_n)_n$ be a sequence of random variables such that there exist $u_n>0$ and $p_n \in \mathbb{R},$ such that $Y_n=\frac{X_n-p_n}{u_n}$ converges in distribution to a non-degenerate random ...
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### Weak law of large numbers for dampened Cauchy distribution

Say I have a sequence of real independent functions $X_i$ on a probability space $(\Omega, P)$ such that $(X_i)_* P$ gives the probability distribution $d\alpha := \frac{cdx}{(1+x^2)\log(1+x^2)}$ (say ...
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### How do I determine $\lim_{n\to \infty}P(|\sum_{i=1}^{n^2}X_i - n^2|\leq n^{\alpha})$ where $(x_i)$ is a sequence of random variables? [closed]

Let $x_1, x_2, \dotsb$ be a sequence of independent random variables distributed uniformly on the interval $[0, 2]$. Find all possible values of the following limit, where $\alpha$ is a positive ...
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### Convergence of the product of two Random Functions

$lim_{t\to\infty}\mathbb{P}(|g(X(t))|>\epsilon)=0$, $f(X(t))=\mathbb{1}_{({X(t)}\neq{0})}$ I want to show $f(X(t))g(X(t))$ coverage in probability to 0. Just wondering if it is possible, thank ...
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### If a probability density function has a limit at infinity, is that limit always zero? [duplicate]

I just read a related question which showed that probability density functions do not always have a limit of zero as a variable approaches infinity because they don't always have a limit at infinity. ...
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Im reading Asymptotic Statistics (Van der Vaart, A. W. (2000)), where in chapter 10 the Bernstein-von-Mises theorem is discussed. So assume we are in the situation where $||P_{(\Theta|X_1,...X_n)} - ... 0answers 11 views ### Converting a trimodal distribution into a uniform distribution Consider some generator$G$that outputs discrete points trimodally distributed along the interval$(0, 1)$, with the points tending to cluster around 0, 1, and 0.5. Then, I need to uniformly ... 1answer 51 views ### Weak convergence of$\mathcal{U}([u_n,v_n])$If$(u_n)_n$and$(v_n)_n$are two sequences of real numbers, such that$\forall n \in \mathbb{N},u_n<v_n,$if$X_n$is a random variable with density$f_{X_n}=\frac{1}{v_n-u_n}1_{[u_n,v_n]},$and ... 1answer 42 views ### Irreducible Markov chain with µ as the stationary distribution. Prove that if$\mu$is a stationary distribution for an irreducible Markov chain on$S$, then$\mu(j) > 0 \ \forall j \in S$. (only using the fact that Markov chain is irreducible and that$\mu P = ...
Assume we have some fixed (finite, may be a density, but not necessarily) function $f(x)$ and some other function $g(\theta,x)$ such that for fixed $x$, $g(\theta,x)$ is in $O(\theta^m)$. ...