# 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.

703 questions
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### Why this sequence converges to $0$ almost surely

I have this sequence of random variables: $$X_{1}(w) = \mathbb{1}_{(1/2,1]}(w), X_{2}(w) = \mathbb{1}_{(0,1/2]}(w), X_{3}(w) = \mathbb{1}_{(3/4,1]}(w), X_{4}(w) = \mathbb{1}_{(1/2,3/4]}(w) ...$$ I ...
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### Convergence of the sum of iid scaled by $n^\alpha$

I am interested in the convergence of the sequence $\mathbb{P}(|X_1+...+X_n|/n^\alpha<z)$ where $z>0$, $\{X_n\}_n$ is an i.i.d. sequence with mean zero and finite variance. I can easily prove ...
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### Lindeberg CLT condition on Discrete Uniform independent sequence of random variables

There is a worked exercise in my book, however there is a line that I am not sure sure about. I understand all of the work before and after this line to finish the proof. Here is what we are given: ...
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### Asymptotic independence on the tail field is equivalent to a probability measure's being trivial

Let $(\Omega, \mathcal B, P)$ be a probability space and $\mathcal A = \bigcap_n \mathcal{A}_n$, where $\mathcal{B} \supset \mathcal{A}_1 \supset ... \supset \mathcal{A}_n \supset ...$ is a decreasing ...
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### necessary and sufficient condition on $(u_n)$ for a.s convergence of $\sum_n u_nX_n$

I want to know if there is a necessary and sufficient condition on the sequence $(u_n)_n$ such that $\sum_n u_nX_n$ converges a.s. wehere $(X_n)_n$ are independent. I know that this series converges ...
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### Expectation of exponential random variables

Can someone help me understand the parts I have highlighted in the proof? I have not understood how $|2c...|$ is upper bounded by $1$ and how one computes the expectation of an exponential random ...
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### Inverse Gaussian Distribution and the Central Limit Theorem

Let the random variables $Y_1,\ldots,Y_n$ be independent and identically distributed (i.i.d.) (standard) Inverse Gaussian random variables with parameters $\mu$ and $\lambda$. Then, let the random ...
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### Limiting Distribution on order statistics

Here is the question: Find the limiting distribution of the quartile ratio defined as $\frac{X_{(3n/4):n}}{X_{(n/4):n}}$ (the third quartile divided by the first quartile)for the exponential ...
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### independent variables implies integrability and same distribution

I want to solve this difficult problem, so I need your help, I want to prove that if $X$ and $Y$ are independent real variables such that $X+Y$ and $X-Y$ are independent, then $X^2$ and $Y^2$ are ...
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### Feller's Proof of the Basic Limit Theorem for Recurrent Events

I'm having trouble understanding the last passage in Feller's proof of the Theorem above, which can be found in section XIII.11 of "An Introduction to Probability Theory and its Applications, vol.1". ...
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I have met with a probability problem which I have no idea to deal with. It says: Let $\alpha>0$, $\beta\geq0$, prove: $\lim_{x\rightarrow+\infty} x^{\alpha+\beta}\mathbb{P}(|X|>x)=0\... 0answers 12 views ### Convergence of row superposition of null array to Poisson point process Background: The limit of superposition of infinite number of independent point processes is a Poisson process under certain conditions. The conditions for the limit process to exist and be Poisson is ... 3answers 151 views ### Limiting a sequence of moment generating functions I was trying to solve the following problem: Let$\{X_n\}_{n=1}^{\infty}$be a sequence of independent random variables with the probability mass function$P\{X_n = \pm1 \} = \frac{1}{2}$,$n \in \...
I have $X_1, X_2, X_3, \cdots$ which are independent random variables with the same non-zero mean ($\mu\ne0$) and same variance $\sigma^2$. I would like to compute \lim_{n\to\infty} P[\frac{1}n\sum^...
Let $X_i$ be i.i.d. sequence of random variables with finite second moment. Then, by the Central limit theorem, $\frac{\sqrt{n}(\bar{X}-E(X))}{\sigma_X} \to N(0,1)$ in distribution. Let $\alpha_n$ ...