# Questions tagged [law-of-large-numbers]

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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### An upper bound to $\mathbb{P}[|\sum_{i=1}^{n}(X_i - \mu)| > \epsilon n]$

Let $\epsilon > 0, \, (X_i)_{i\geq1}$ be a sequence of identically distributed (we denote $EX_i$ by $\mu$) and pairwise (not mutually!) independent random variables prove that there exists some ...
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### law of large numbers against several distributions

Suppose there are $K$ distributions $F_1,..,F_K$ and a random variable $x$ such that $E(x | F_i) = \mu_i$. Suppose also that the distributions appear in proportion $p_1 F_1 + ... + p_n F_K$, so that ...
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### Law of large numbers on a random number of samples

Let $X_1,\dots,X_n$ be $n$ iid random variables, each with expectation $1$. From the law of large numbers one has $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n X_i = 1 \text{ a.s.}$$ Now let $Y_n$ be a ...
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### Feller's probability theory problem 5 from Ch. X - Conditions for LLN and CLT

I am Reading Feller's Introduction to Probability Theory and Its Applications and I found an exercise from Chapter 10 difficult to solve. In this exercise (which is the number 5 in the Intl. Edition) ...