# 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).

511 questions
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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 ...
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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: ...
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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 ...
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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 $X_n$ ...
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### Uniform Law of large numbers

Could you please help with the proof of the proposition: Let $G(\cdot)$ be bounded, continuous, strictly increasing function on $\mathbb{R}.$ Let ${\xi_t},\, t\in\mathbb{Z}_+$ be i.i.d random ...
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### Strong law of large numbers for Poisson process

My question regards the strong law of large numbers as stated, e.g., in Ethier and Kurtz (1986, p. 456 Eq. (2.5)), as follows: If $Y$ is a unit Poisson process, then for each $u_0>0$, \begin{...
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### Find the limit of $\sum\limits_{r=\lfloor an \rfloor}^{\lfloor bn \rfloor} {n \choose r } p^r (1-p)^{n-r}$ using the central limit theorem

Let $p \in(0,1)$. What is the distribution of the sum of $n$ independent Bernoulli random variables with parameter $p$? Let $0 \leq a < b \leq 1$. Use approprtiate limit theorems to determine how ...
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### S.N. Bernstein Law of Large Numbers

while reading a paper named "Network Embedding as Matrix Factorization: UnifyingDeepWalk, LINE, PTE, and node2vec" (http://keg.cs.tsinghua.edu.cn/jietang/publications/WSDM18-Qiu-et-al-NetMF-network-...
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### Find the limit of $\int_0^1\cdots\int_0^1 \sin(\sqrt[n]{x_1\cdots x_n}) \, dx_1\cdots dx_n$ using a probabilistic approach

I am looking for the solution of this: $$\lim_{n \rightarrow \infty} \int_0^1\cdots\int_0^1 \sin(\sqrt[n]{x_1\cdots x_2}) \, dx_1\cdots dx_n$$ I know that $X_1,\ldots,X_n \sim \mathcal{U}[0,1]$....
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### Does this sequence of random variables satisfy the Law of Large Numbers?

Notation: $S_n = X_1 + \ldots + X_n$ Definition $1$ : The sequence of random variables $X_1, X_2, \ldots$ satisfies the Weak Law of Large Numbers if $$\frac{S_n - E[S_n]}{n} \overset{p}{\to} 0$$ ...
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### Finiteness of the sum of the product of an i.i.d. sequence

Before I go to the statement of my question I just want to say a few words about the personal background of this question. I have recently taken a course on stochastic differential equations without ...
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