# Questions tagged [convergence]

Convergence generally means that a sequence of a certain sample quantity approaches a constant as the sample size tends to infinity. Convergence is also a property of an iterative algorithm to stabilize on some aim value.

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### Proving the convergence of the maximum of Uniform Distribution

I have a random sample of size $X_1, X_2, .., X_n$ following $U(0,2)$. I need to prove that $X_{(n)}$ which is the maximum ordered statistics will converge to $2$ in probability and almost surely. I ...
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### MCMC beginner question at an example chain plot: Do I need more steps? How much burn-in do I need, if I can tell already?

I am using the emcee python library to fit a model to data via MCMC. Below an example plot for the chain of one of my parameters. Here I ran 1000 steps with 100 walkers. Now I have two beginner ...
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### Does the MLE converge in mean-square?

Simple question: does the MLE of a (finite-dimensional) parameter converge in mean square to the true value, that is, $$\mathbb E[\Vert\hat\theta_\text{MLE} - \theta\Vert^2_2]\rightarrow 0.$$ I know ...
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### Consistency of a simple Bayes classifier

Let $p(x,y)$ be the joint distribution of random variables $X$ and $Y$ where \begin{aligned} Y&\sim \operatorname{Bernoulli}(\pi),\\ X\mid Y=y&\sim N(\mu_y,\sigma_y^2). \end{aligned} Let ...
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### $\limsup$ in proof that $X_n = o_p(Y_n)$ and $Y_n = O_p(1)$ then $X_n = o_p(1)$

In the proof where we have $$P(|X_n| \geq \varepsilon) \leq P\left(|\frac{X_n}{Y_n}| \geq \frac{\varepsilon}{B}\right) + P(|Y_n| > B)$$ why do we need to take the $\limsup$ to show $X_n = o_p(1)$ ...
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### What is the limiting distribution of $\chi_r^2$ random variable, where $r\to 0^+$

What is the limiting distribution of $\chi_r^2$(Chi-square) random variable, where $r\to 0^+$. The following picture shows that as $r\to 0^+$ the distribution become degenerated in zero point. If it ...
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### Would very small stratum be a problem for ANCOVA model?

I know that for logistic regression, if you get an empty cell, the model may not run at all. How about continuous outcome? If one of the categorical predictors has very small stratum, would there be a ...
Say that we have $\sqrt{n}(\hat{\mu} - \mu_0)$, which we can equivalently write as $\sqrt{n}(\hat{\mu} - \mu) + \sqrt{n}(\mu - \mu_0)$, where $\mu$ is the population mean, $\hat{\mu}$ is the sample ...