# 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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### Monotonicity of the probability of a sum of independent random variables being below a threshold

Suppose I have a sequence of i.i.d. random variables $X_1, X_2, \dots$ with positive mean $E[X_1] = \mu$. Is the function $$f(n) = \Pr( X_1 + X_2 + \dots + X_n \leq A)$$ monotonically non-increasing ...
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### Central Limit Theorem for Difference-in-Means Estimator

I am studying Lecture 1 of Stefan Wager's Causal Inference notes and come across a central limit theorem for the difference-in-means estimator, which I am unable to prove. The mathematical abstraction ...
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### Convergence of a capital describing random variable

In my previous question Infinite game of an unfair coin toss, I showed that in a game between Person A and B, where: Person A has an unfair coin with probability $p \in (\frac{1}{3},\frac{1}{2})$ of ...
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### sum of two point distribution converge to infinity almost surely

Question: $\{X_k\}_{k=1}^{\infty}$ i.i.d.,$P(X_1=-1)=P(X_1=1)=\frac12$. $S_n=\sum\limits_{k=1}^n X_k$. Prove that: 1.$S_n$ converge to infinity almost surely,i.e. $P(\lim_{n\to+\infty}S_n=\infty)=1$; ...
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### Weak vs strong law of large numbers

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of real i.i.d. random variables with mean $\mu$. Let $S_n$ be the sum of the first $n$ elements of this seqeuence, $$S_n = \frac1n\sum_{i=1}^n X_i.$$ ...
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### Convergence in Probability (Law of Large Numbers)

Suppose $X_1,X_2,…,X_n$ are iid Poisson random variables, each with mean $\theta$. How to prove that $Y_n=\exp[−\frac{1}{n}(X_1+X_2+⋯+X_n)]$ converges in probability to $P(X=0)=\exp(−\theta)$ ? Hint: ...
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### Convergence in Probability (Weak Law of Large Number)

Suppose $𝑌_1, 𝑌_2, … , 𝑌_𝑛$ are independent and identically distributed Poisson random variables, each with mean $𝜆$. Prove that $𝑋𝑛 =exp[−(1/𝑛)(𝑌_1 + 𝑌_2 + ⋯ + 𝑌_𝑛)]$ converges in ...
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### Will the expanding cube look gray?

Consider the tiling of $\mathbb{R}^n$ by unit cubes centered at integer lattice points, i.e. $$\mathbb{R}^n = \bigcup_{a \in \mathbb{Z}^n} Q\left(a, \frac{1}{2}\right).$$ Color each unit cube black ...
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### Strong Law of Large Numbers for increasing index sets

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of integrable i.i.d random variables. Let $I_n \subset \mathbb{N}$ satisfy $\lim_{n \to \infty} |I_n| = \infty$, where $|I_n|$ is the cardinality of $I_n$. ...
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### Why are these definitions of the Law of large numbers the same?

I have learnt the following definition of the law of large numbers: Theorem 17.4 (Law of Large Numbers). Let $X_1, X_2, \ldots$, be a sequence of independent and identically distributed random ...
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### Polya's urn, should I use martingales or LLN

I am trying to prove the following question, but I am finding it a bit tricky to determine the distribution of $X_i$ (the number of red balls drawn in the $i$-th round) and thus I do not know which ...
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Suppose $U_1,...,U_n$ is a simple sampling from a uniform distribution $U(0,1)$ and $G_n(u)$ is an empirical distribution function. Prove that  \begin{gathered} n \int_0^1\left(G_n(s)-s\right)^2 d s=...