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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).

The law of large number (lln) describes what happens if we perform an experiment a large number of times. It states that the average of the results obtained from a lot of trials will be close to the expected value. It also states that it will get closer to the expected value as more trials take place. The law guarantees a stable long-term results for the averages of events.

The strong law of large numbers states that the averages converge a.s., to the expected value \begin{equation*} \overline{X}_n\to \mu,~ n\to\infty . \end{equation*}

The weak law of large numbers states that the average converges in probability towards the expected value: \begin{equation*} \lim_{n\to\infty}Pr(|\overline{X}_n-\mu|>\epsilon)=0. \end{equation*}

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What is the difference between the weak and strong law of large numbers?

I don't really understand exactly what the difference between the weak and strong law of large numbers is. The weak law says \begin{align*} \lim_{n \rightarrow \infty} \mathbb{P}[\mid \bar{X}_n - \mu \mid \leq \epsilon ] = 1, \end{align*} while the…
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Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the results from multiple trials will tend to converge…
vantage5353
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Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance

I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement: Let $(X_n)$ be a sequence of dependent…
Elements
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Gambler's fallacy and the Law of large numbers

Can someone explain me, how the Law of large numbers and the Gambler's Fallacy do not contradict. The Gambler's Fallacy says, that there is no memory in randomness and any sequence of events has the same probability as any other sequence. However,…
clamp
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What does a probability of $1$ mean?

From a textbook on probability on the Law of Large Numbers: Theorem 3-19 (Law of Large Numbers): Let $X_1,X_2, \ldots , X_n$ be mutually independent random variables (discrete or continuous), each having finite mean and variance. Then if $S_n =…
curryage
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The strong and weak laws of large numbers: Why two?

The following questions are entirely based on the corresponding article from Wikipedia. The assumptions of both laws are the same, and the strong law has a more general claim than the one of the weak law. The question is then: what is the reason for…
Ivan
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What we can deduce from $\frac{S_n}{n}\rightarrow 0$ in probability?

Let $(X_n)$ be a sequence of independent random variables with distribution $$\mathbb{P}(X_n = -n) = \mathbb{P}(X_n = n) = \frac{1}{2} p_n,\mathbb{P}(X_n = 0) = 1 - p_n$$ where $0 \leq p_n \leq 1$. Prove that $\dfrac{S_n}{n}$ converges in…
Alex Nguyen
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Sum of $X_k$ with $\mathbb{P}(X_k = \pm1) = \frac12 \pm \frac{1}{k}$ independently

Let $\{X_k\}$ be a sequence of mutually independent random variables with \begin{align} \mathbb{P}(X_k = 1) & = 1/2 + 1/k, \\ \mathbb{P}(X_k = -1) & = 1/2 - 1/k \end{align} for each $k \ge 1$. Question a) What is $\mathbb{P}(\sum_{k=1}^\infty X_k =…
Nuno
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Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)\log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ Prove that $X_n$ satisfies weak law of large numbers but doesn't…
nilcorc
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Law of large numbers for dependent random variables with fixed covariance

Given identically distributed random variables $X_1,...,X_n$, where $ \mathbb E X_i = 0$, $\mathbb E [X_i^2] = 1$, and $\mathbb E [X_i X_j] = c<1$, define $S_n = X_1 + ... + X_n$. Will $$\frac{S_n}{n} \rightarrow 0$$ in probability as $n\rightarrow…
The_Anomaly
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Help understanding the weak law of large numbers with respect to statistics

I'm trying to do some self-studying to upgrade my statistics knowledge, and came across this term in a section discussing the weak law of large numbers and Bernoulli's theorem: $$\sum_{k=0}^n k\frac{n!}{k!\,(n-k)!}p^k(1-p)^{n-k}$$ According to the…
PizzAzzra
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How to show Law of Large numbers for this random walk?

Let $X_{0}=1$ and define the Markov Chain $X_{n}$ given by the transition probabilities $p_{01}=1$ $p_{k,k+1}=p$ and $p_{k,k-1}=(1-p)=q$, $k\geq 1$ where $p$ is some fixed number in $(0,1)$. I want to show that if $p\leq\frac{1}{2}$, then…
Dovahkiin
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Result analogous to the Central Limit Theorem if the third moment is also finite

Motivation Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables that have finite first moment. Let $S_n =\sum_{i=1}^n X_i$. We have the Law of Large Number $$ n^{-1}S_n \to \mathbb{E}[X_1] \quad \text{a.s.} $$ We can view…
Tututuu
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SLLN when the expectation in infinite

In a Post I found it says: Whenever ${\rm E}(X)$ exists (finite or infinite), the strong law of large numbers holds. That is, if $X_1,X_2,\ldots$ is a sequence of i.i.d. random variables with finite or infinite expectation, letting $S_n =…
Lambda87
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Weighted strong law of large numbers

Let $\{X_n\}_n$ be a sequence of i.i.d. (discrete) random variables with expected value $\mu$, and let $\{B_n\}_n$ be a sequence of finite and bounded random variables. Assume that for all $n$, $B_n$ is independent from $X_n$ but not necessarily…
Lios
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