# 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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I am studying probability theory on my own and am trying to work the following problem in the book - Let $X_1, X_2, . . .$ be independent, $U(0, 1)$-distributed random variables, and let $Nm \in Po(m)... 1answer 2k views ### Markov chain convergence, total variation and KL divergence I have a few related questions regarding the convergence of continuous-state Markov chains. The theorems that I found claim that Markov chains converge in total variation if they are$\phi$-... 2answers 3k views ### Convergence to a Uniform Distribution$\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$. ... 1answer 148 views ### Can we go from$X_n = \mu + O_p(n^{-1})$to$E[X_n] = \mu + O(n^{-1})$? Let$X_n$be a uniformly integrable (UI) sequence of random variables. If we have $$X_n = \mu + O_p(n^{-1}),$$ then for$0 \le \delta < 1$this implies $$X_n = \mu + o_p(n^{-\delta}) \quad \quad ... 2answers 817 views ### Limit of a convolution and sum of distribution functions I need to prove an induction step. X_i are independently distributed with the distribution function 1-F_i=x^{-\alpha}L_{i}(x) where \alpha \geq 0 and L_{i}(x) is regularly varying (If the ... 3answers 1k views ### Prove that this doesn't converge almost sure to 0 Suppose we have X_n a random variable, that can take two values: X_n = \begin{cases} 0, & \text{with probability 1 - \frac{1}{2n},} \\ n, & \text{with probability \frac{1}{2n}} \end{... 1answer 333 views ### Convergence of the Matérn covariance function to the squared exponential The Matérn covariance function converges to the squared exponential covariance function. Many sources, amongst them the GPML book and Wikipedia, state this result. None of them provide details. I ... 2answers 205 views ### Chebychev’s Weak Law of Large Numbers This theorem is on Econometric Analysis (7th edition) by Greene (2012), Page 1071. It states that "If x_i, i=1,2,...,n is a sample of observations such that E(x_i)=\mu_i<\infty and var(x_i)=\... 1answer 3k views ### Convergence of identically distributed normal random variables I had this example in my machine learning lecture. Let X_2,\ldots,X_n be identically distributed (but not independent) copies of X_1 drawn from \mathcal N(0,1). Then X_n converges to Y = -... 1answer 102 views ### Rate of convergence of \hat Q_{xx}^{-1} = \left(\frac{\mathbf{X}^T \mathbf{X}}{n}\right)^{-1} to the probability limit? Consider the simple linear regression model.$$ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad \quad \quad \quad i = 1,2,\dots,n. $$Let \mu_x and \sigma_x^2 represent the mean and variance of ... 1answer 758 views ### MCMC convergence, analytic derivations, Monte Carlo error I'm trying to figure out some convergence statements on an MCMC example. The setup is: I'm generating data samples as observations from a (known) deterministic parameter, say s (using a forward ... 1answer 455 views ### Convergence issues with lme4 1.1-20 for models that converged when using earlier version of lme4 I am encountering convergence problems with some models after updating to lme4 1.1-20 that I did not encounter with earlier versions of lme4 (in particular, lme4 1.1-15). I am encountering these new ... 0answers 44 views ### Convergence in Distribution in Order Statistics Let X_1, X_2, \ldots be iid from Exp(\theta) with density function f(x) = \frac{1}{\theta}e^{-\frac{x}{\theta}}. (a) Find the limiting distribution of M_n = Y_1 - \theta\ln(n) and T_n = nY_n,... 3answers 1k views ### A dynamical systems view of the Central Limit Theorem? (Originally posted on MSE.) I have seen many heuristic discussions of the classical central limit theorem speak of the normal distribution (or any of the stable distributions) as an "attractor" in ... 1answer 2k views ### Why does MAP converge to MLE? In Kevin Murphy's "Machine learning: A probabilistic perspective", chapter 3.2, the author demonstrates Bayesian concept learning on an example called "number game": After observing N samples from ... 1answer 2k views ### Deriving K-means algorithm as a limit of Expectation Maximization for Gaussian Mixtures Christopher Bishop defines the expected value of the complete-data log likelihood function (i.e. assuming that we are given both the observable data X as well as the latent data Z) as follows:$$ \... 2answers 2k views ### 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 ... 2answers 666 views ### 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$n \rightarrow \infty$? (... 2answers 4k views ### 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 ... 2answers 3k views ### How do I interpret my validation and training loss curve if there is a large difference between the two which closes in sharply my CNN is meant to classify an image as one out of around 30 categories. I am training on 6400 samples using a batch size of 128. I am using Keras/ Tensorflow Architecture is Conv + Batch ... 1answer 1k views ### 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 ... 3answers 212 views ### When do posteriors converge to a point mass? What are the necessary conditions for a model's posterior to converge to a point mass in the limit of infinite observations? What is an example that breaks this convergence result? Off the top of my ... 2answers 437 views ### Econometrics text claims that convergence in distribution implies convergence in moments The following lemma can be found in Hayashi's Econometrics: Lemma 2.1 (convergence in distribution and in moments): Let$\alpha_{sn}$be the$s$-th moment of$z_{n}$, and$\lim_{n\to\infty}\alpha_{sn}...
It's clear to me by inspection that if we fix $\beta = \frac{1-\mu}{\mu} \alpha$ (thereby fixing the mean) and let $\alpha \rightarrow 0$, the Beta distribution approaches a Bernoulli($\mu$) ...