Questions tagged [law-of-large-numbers]
Several theorems stating that sample mean converges to the expected value as $n\to\infty$. There is a weak law and a strong law of large numbers.
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Can we have $|\overline{X}_n - E[X_1]| > \varepsilon$ infinitely often?
Let's say I have some random variables, $X_1, X_2, ...$ which are identically distributed with finite expected value, $E[X_1]$ and say they satisfy the requirements of the law of large numbers. Let
$$\...
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Almost sure convergence in Bayesian setting
Lets say I have a probability space with random variables X1,X2,.... These random variables have a parameter Θ. Given Θ, X1,X2,... are iid. This implies that conditional on Θ, the sample mean ...
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How to prove that support converges to probability?
In the body literature of Association Rule Mining (apriori algorithm is one of them) there's a lot of information about te usage of many metrics, whithin them 'support'.
Support is defined as the ...
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Sum of asymptotically independent random variables - Convergence
Let $\theta_N=\frac{1}{N}\sum_{i=1}^N \pi_i\cdot g_i$ where $0<\pi_i<1$ and $0<g_i<1/\pi_i$ such that $\theta_N\overset{N\rightarrow \infty}{\rightarrow}\theta$.
If $X_i\sim Ber(\pi_i)$, I ...
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Weak Law of Large Numbers: Conditional Expectations in Random Subsequences
Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}\subset\mathbb{R}$ be decreasing ...
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Almost sure convergence of $\frac{2}{n(n-1)}\sum\limits_{1 \leq i < j \leq n} X_i X_j$
I'm trying to prove that:
Given a sequence $(X_n)_{n \geq 1}$ of independent and identically distributed random variables, $E(X_i^2) < +\infty$ for all $i \geq 1$, then $$\frac{2}{n(n-1)}\sum\...
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Does the Law of Large Numbers work better for some Distributions? [closed]
Here are two popular principles in Statistics:
1) Law of Large Numbers: If $X$ is a random variable with a probability density function $f(x)$ and an expected value $E[X] = \mu$. If we take a sample ...
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Timeseries problem with law of large numbers
Let us have an AR(1) model with individual efect
$$y_t = \alpha + \theta y_{t-1} + \varepsilon_t$$
with $|\theta|<1$ for stacionarity and $\varepsilon_i$ i.i.d. from distribution with mean $0$ and ...
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When is a function of an ergodic stationary process itself ergodic stationary?
I am working with a function which has the form $f(X_1, \dots, X_n)$, where $\\{X_n\\}$ is an ergodic stationary process. Theorem 5.6 in "A first course in stochastic processes" by Karlin &...
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Convergence in probability and boundness in probability with respect to sample mean and sample variance
This is a question about the convergence in probability and boundness in probability.
Suppose $X_i \overset{\textrm{i.i.d.}}{\sim} (\mu, \sigma^2 )$ for $i=1,2, \cdots, n$.
Denote $\overline{X}$ and $\...
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Show that $\frac{1}{n(n-1)} \sum_{i\neq j} \sin\left(X_i X_j\right)$ converges almost surely to a constant
Let $X_i$ be iid random variables. How does one show that
$$ \frac{1}{n(n-1)}\sum_{i\neq j}^n \sin\left(X_i X_j\right) $$ converges almost surely to a constant?
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Large sample distributions
Suppose we have observations $x_1, x_2, \ldots, x_n$ where $n$ is very large. Now we standardize the observations as $$y_i=\frac{x_i-\bar{x}}{\frac{s}{\sqrt{n}}},$$ where $s=\frac{\sum\limits_{i=1}^n(...
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Convergence of estimated Survival Functions
Q1 part A&B I have so far
$$\underset{n\rightarrow\infty} {\lim} \frac{1}{n}\sum_{i=1}^nI(T_i>x)$$
since we are summing an indicator variable we can say it has a Bernoulli distribution with ...
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Is the Law of Large Numbers Related to the Occurrence of Rare Events Over Many Trials?
I recently watched an episode of "The Big Bang Theory" where Sheldon makes a comment about the Law of Large Numbers. In the episode, Sheldon realizes he needs eggs, and almost immediately ...
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Probability Theory. Moving from finite to continuous
This has probably been asked before, as this is (I think) a fundamental theory of statistical theory, but I don't know what it is called, hence I have not yet found an answer.
Consider a box which ...
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