# Tagged Questions

For questions about the law of large numbers, a classical limit theorem in probability about the asymptotic behavior (in almost sure or in probability) of the average of random variables. To be used with the tags (tag:probability-theory) and (tag: limit-theorems).

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### Law of large numbers for positive random variable [closed]

can someone pls help me with this problem: $X_{n}$ iid with expectation $\infty$ and $X_{n} \ge 0$ then: $S_{n}/n \to \infty$ a.s.
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### Can Luck Be Proven Mathematically?

Here is what I'm thinking, and it has to do with the Gambler's Fallacy and Law of Large Number. The Gambler's Fallacy states that due to the probability of an event is statistically independent and ...
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### Does this context satisfy the hypothesis of the law of large numbers?

A precondition for the law of large numbers is that $X$ is a random variable with $X_1$, $X_2$, $\ldots$ being a sequence of i.i.d. random variables s.t. $E[X] = E[X_i]$. Now suppose $X$ is the ...
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### How is this not a counter-example to the law of large numbers?

Let $\Omega = \{0,1\}$ and $X: \Omega \rightarrow \{0,1\}$ be a random variable s.t. $X = id$ with $E[X] = 0.5$ (i.e., $P(0) = 0.5 = P(1)$). Let $X_1$, $X_2$, $\ldots$ be a sequence of i.i.d. random ...
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### The meaning of “EXACT laws of large numbers”

I have come across various papers that consider a stronger form of probability-relative frequency convergence theorem called the 'exact law of large numbers". I note that in particular such theorems ...
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### Calculating Number of possible solutions to large Integers [closed]

I was looking for the best possible mean to determine all possible integer solution sets for large number such 21527411027188897018960152013128254292577735888456 ...
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### Application: Law of Large Numbers

We have two collections of random variables $X_i$ and $Y_i$. The $X_i$ are independent and identically distributed with expectation $1$, and the $Y_i$ are also independent and identically distributed ...
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### Law of Large Numbers for the reciprocal

Assume $(X_i)_{i\geq1}$ and $(Y_i)_{i\geq1}$ are two independent sequences of i.i.d. random variables such that $\mathbb E[X^k]<\infty$ and $\mathbb E[Y^k]<\infty$ $\forall k\geq1$. I am ...
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### Proof of strong law of large numbers: not identical case

I've seen two types of conditions for strong law of large numbers: one requires i.i.d and first order moment condition: $X_n$ i.i.d with $E|X_1| < \infty$; the other requires second order moment ...
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### Almost sure convergence SLLN practice example

Incidents for product $A$ occur at time $T_1, T_2,\dots$ where $T_i=X_1+X_2+\dots+X_i$. Assume that $(X_i)_i$ are i.i.d. and let $M(t)=\max\{n:T_n \leq t\}$ the number of incidents occured at time $t$....
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### If a class of functions $\mathcal{F}$ is a Glivenko-Cantelli class then it is also a Donsker class?

Definitions: Consider a random variable $X:\Omega \rightarrow \mathcal{X}$ defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ with probability distribution $P$. All functions ...
### The bracketing number and entropy: how do they vary with $\epsilon$ and $r$?
I have some doubts related to the bracketing number and entropy as defined in van der Vaart "Asymptotic Statistics" p.270. Definitions: Consider a random variable $X:\Omega \rightarrow \mathcal{X}$ ...