# 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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### $X_i = i^{\delta}Z_i$, find the range of $\delta$ for which the law of large numbers and the central limit theorem are valid

$Z_1,Z_2,...$ are i.i.d., their expected value is zero, their variance $\sigma^2$, and $E[|Z_i^2|] = m_3 < \infty$. $X_i = i^{\delta}Z_i$, find the range of $\delta$ for which the law of large ...
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### Question about book solution to estimate $E[e^{XY}]$ when $X$ and $Y$ are independent exponential RVs with $\lambda = 1$

Let $X$ and $Y$ be independent exponential random variables with mean 1. (a) Explain how we could use simulation to estimate $E[e^{XY}]$. (b) Show how to improve the estimation approach in part (a) by ...
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Suppose that $X_1,X_2,\cdots,X_n$ are iid sequence with pdf $\frac{2}{\pi (1+x^2)}\cdot 1_{(0,+\infty)}(x)$. Denote $S_n$ as $S_n:=X_1+X_2+\cdots+X_n$. Prove that there exits $c>0$ such that $$\... 1answer 143 views ### Law of large numbers and theoretical probability I didn't exactly know how to phrase the title of this question so a little more information.. I was conducting a small experiment with a class of secondary-school students to demonstrate the law of ... 1answer 27 views ### Expectation of the mean of the sum of random variables [closed] If X_i's are independent and identified random variables, each with mean \mu and variance \sigma^2. Let's say S_m = \frac{1}{m} \sum_{i=1}^m X_i,~~ m = 1,2,\ldots,M. What are the values of \... 2answers 68 views ### With X \sim Unif(0,1) what is the limit of \frac{n}{x_1^{-1} + \cdots + x_n^{-1}} I am confused as to how I can tackle this question: With X \sim Unif(0,1) what is the limit of \frac{n}{x_1^{-1} + \cdots + x_n^{-1}}. My assumption is that is 0. but I would like to show that ... 0answers 56 views ### If \lim_{n \rightarrow \infty} \frac{S_n^4}{n^4} = 0 then \lim_{n \rightarrow \infty} \frac{S_n}{n} = 0 If \lim_{n \rightarrow \infty} \frac{S_n^4}{n^4} = 0 then \lim_{n \rightarrow \infty} \frac{S_n}{n} = 0 where S_n is the sum of n iid RVs with mean zero. My question I'm having trouble ... 2answers 104 views ### Central Limit Theorem for geometric mean Suppose that X_1,X_2,... be i.i.d. variable uniformly distributed on (0,1), and let \tilde{X_n} denote the geometric average of n of these variables, i.e.: \tilde{X_n}=(X_1X_2\cdots X_n)^{1/n}.... 1answer 30 views ### Almsot surely convergence of series (a) Suppose that X_1,X_2,... be independent with P(X_n=n-1)=\frac{1}{n}, P(X_n=-1)=1-\frac{1}{n}. Show that there are no constants {\mu_n} such that \frac{s_n}{n}-\mu_n \rightarrow 0 a.s. (... 0answers 27 views ### Question about convergence of a series - need help understanding a proof for the strong law of large numbers My question: In the proof below for the strong LLNs, I don't know how the author goes from the \color{red}{\text{red box}} to the \color{blue}{\text{blue box}} (see screenshot below) Getting to ... 1answer 42 views ### Question about the strong law of large numbers (to build understanding) I'm learning about the LLN and CLT for the first time and I'm having some trouble. I've read through other posts but I have a quirky (likely dumb) question about denominators... The Strong LLN ... 0answers 70 views ### A Problem on the Limit of an Integral While I investigate the property of positive random variables, I encountered the following question, not easy to solve. Let f:\left[0,\infty\right)\rightarrow\mathbb{R} be a continuous positive ... 0answers 54 views ### A Problem on the Limit of a Sequence While I investigate the property of positive random variables, I encountered the following question, not easy to solve. Let \left\{ a_n \right\}_{n=1}^{\infty} be a sequence that satisfies a_n &... 1answer 70 views ### Does variance of sample mean converge to zero? n random variables X_1,\ldots,X_n are an i.i.d. sample. \bar X_n is the sample mean. \mu is the expectation of distribution. Doesn't guarantee a finite variance. Does this always hold?$$E[(\...
I'm trying to work my way through a problem which defines $N_t$ as a Poisson process of rate $\lambda$ and $X_n = N_n − n,\quad\text{for }\; n = 0, 1, 2, \ldots$ I've explained why $X_n$ is a ...