# 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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### Debunking wrong CLT statement

The central limit theorem (CLT) gives some nice properties about converging to a normal distribution. Prior to studying statistics formally, I was under the extremely wrong impression that the CLT ...
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### Convergence in distribution, probability, and 2nd mean

Let $\mathbb P(X=1) = \mathbb P(X=-1) = 1/2$. Define $$X_n = \begin {cases} X & \text{with probability } 1- \frac{1}{n}\\ e^n & \text{with probability } \frac{1}{n} \end {cases}$$ ...
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### Limiting distribution of the first order statistic of a general distribution

Let $Z_i,Z_2,\ldots$ be IID Random Variables with density $f$. Suppose that $P(Z_i>0)=1$ and that $\lambda=\lim_{x \to 0+} f(x)>0$. How can I show that $X_n=n \times \min\{Z_i\}$ has a limiting ...
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### Markov chain convergence, total variation and KL divergence

I have a few related questions regarding the convergence of continuous-state Markov chains. The theorems that I found claim that Markov chains converge in total variation if they are $\phi$-...
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### Convergence to a Uniform Distribution

$\newcommand{\floor}{\left\lfloor #1 \right\rfloor}$ Show that if $P(X_n = i/n)=1/n$ for every $i = 1,...,n$, then $X_n$ converges in distribution to a uniformly distributed random variable $X$. ...
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### Does a Binomial converge to Poisson or Normal?

I have read the answer here. Here the distinction is that If $n\to\infty$ and $p\to0$ while $np$ approaches some positive number $\lambda,$ then the binomial distribution approaches a Poisson ...
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### Optimizing OLS with Newton's Method

Can ordinary least squares regression be solved with Newton's method? If so, how many steps would be required to achieve convergence? I know that Newton's method works on twice differentiable ...
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### What happens to the likelihood ratio as more and more data is gathered?

Let $f$, $g$ and $h$ be densities and suppose you have $x_i \sim h$, $i \in \mathbb{N}$. What happens to the likelihood ratio $$\prod_{i=1}^n \frac{f(x_i)}{g(x_i)}$$ as $n \rightarrow \infty$ ? (...
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### High-dimensional regression: why is $\log p/n$ special?

I am trying to read up on the research in the area of high-dimensional regression; when $p$ is larger than $n$, that is, $p >> n$. It seems like the term $\log p/n$ appears often in terms of ...
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### Understanding the concept of "Bounded in probability"

My statistics book defines the concept of "bounded in probability" in the following way: Definition 5.2.2 (Bounded in Probability). We say that the sequence of random variables $\{X_n\}$ is ...
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### Is Slutsky's theorem still valid when two sequences both converge to a non-degenerate random variable?

I am confused about some details about Slutsky's theorem: Let $\{X_n\}$, $\{Y_n\}$ be two sequences of scalar/vector/matrix random elements. If $X_n$ converges in distribution to a random ...
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### What are some reasons iteratively reweighted least squares would not converge when used for logistic regression?

I've been using the glm.fit function in R to fit parameters to a logistic regression model. By default, glm.fit uses iteratively reweighted least squares to fit the parameters. What are some reasons ...