Convergence generally means that a sequence of a certain sample quantity approaches a constant as the sample size tends to infinity.

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Problem of infinite or missing values in Hessian at convergence in the case of joint longitudinal and survival sub model

I'm estimating a Joint longitudinal and survival model using the JM package. The procedure works well for a subset of my data, but if I try to use the entire data set I get the following warning ...
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27 views

convergence of geometric mean/harmonic mean

Does any one know papers regarding the convergence of geometric mean or harmonic mean in probability, parallel to central limit theorem?
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43 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 $\frac{1}{2n}$} ...
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9 views

How can convergence (in distribution) be assessed in the context of multiple imputation by chained equations?

The MICE algorithm starts by randomly imputing the missing values in a dataset, and then proceeds to predict the missing values in each variable by modeling the relationship between the non-missing ...
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4 views

Convergence issues: maximal model converges, model with fewer predictos won't

I've run the below model in lme4 (lmer) without any issue: ...
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17 views

Why does glmer break when I remove a subject?

I'm working with the epilepsy data set from Applied Longitudinal Analysis by Fitzmaurice et al. (http://www.hsph.harvard.edu/fitzmaur/ala/epilepsy.txt). In this trial, 59 patients are split into a ...
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23 views

Example when $\hat{R}$ diagnostic is failing [duplicate]

I've seen somewhere an example of $\hat{R}$ statistics being close to 1 and the chains not converging. Marc Kery brought up this example: where the chains still converge but they all do not ...
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1answer
24 views

Convergence in Distribution for a sequence of standardized chi square random variables

Let $\{X_n\}$ be a sequence of random variables where $X_n \sim \chi ^2_{_n} \forall n $ . The sequence $\{X_n\}$ has an associated sequence of MGFs given by $\{M_{x_n}(t)\}$ ,where $ ...
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48 views

Does the product of three Gaussian random matrices converge in distribution to a Gaussian?

Suppose we have vectors $u,v \in \mathbb{R}^r$ with and matrix $W \in \mathbb{R}^{r \times r}$ where all entries of $u,v,W$ are iid $N(0,1)$. Does the following hold? \begin{equation} \frac{1}{r} ...
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1answer
61 views

ergodic theory for markov processes

For an ergodic Markov Chain $$ \frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f] $$ where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
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16 views

Definition of Stable convergence in law: why do we need an extension of the probability space?

I am trying to understand the definition of stable convergence in law. I have found the following definition. Definition. Let $Y_n$be a sequence of random variables defined on a probability space ...
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55 views

Markov model parameter concentration and Fisher Information Matrix

For iid data, the posterior on the parameter $$ p(\theta \mid x_{0:T}) = \prod_{t=0}^T p(x_t \mid \theta) p(\theta) $$ is known to become independent of the prior which is the Bernstein-von Mises ...
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2answers
107 views

Independence of reward and future state in stochastic process?

Consider a Markov decision process in which we transition from state $s_t \rightarrow s_{t+1}$ by taking action $a_t$, and then apply an update to a single entry from a table of $Q$-values based on a ...
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6 views

convergence of coordinate descent applied to lasso

When using coordinate descent for solving a lasso regression, does normalizing the features impact the convergence rate?
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2answers
162 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 ...
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2answers
50 views

What happens to integration over a term that converges to zero in probability?

I have to do integration like this $\int h(x) [\hat{g}_n(x) - g(x)] dx$ ,where $\hat{g}_n(x)$ is a non-parametric estimator of $g(x)$ and $\hat{g}_n(x) - g(x) = o_p(1)$; $h(x)$ is an arbitrary ...
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1answer
55 views

Convergence diagnostic of Markov chain that converge to uniform

Let $\Omega$ be a finite state space, $(X_t)_{t\in\mathbb{N}}$ be a discrete-time Markov chain that converges to the uniform distribution, and $P$ be its transition matrix. I'm looking for different ...
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1answer
87 views

Simple example wanted: $ X_n $ converges to $X$ in probability but not almost surely

I'm looking for a simple example sequence $\{X_n\}$ that converges in probability but not almost surely. The example I have right now is Exercise 47 (1.116) from Shao: $ X_n(w) = \begin{cases}1 ...
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197 views

Convergence in distribution of the following sequence of random variables

$X_n\sim Beta\left(\frac{\alpha}{n},\frac{\beta}{n}\right)$ with $\alpha>0$ and $\beta>0$. Does $X_n$ converges to a distribution? How do I approach to show that this converges to a ...
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0answers
22 views

Asymptotic normality and rate of convergence of the mean

Is there any results on the rate of the convergence of the mean of a random variable being asymptotically normally distributed. For example, let $X_n$ be such that $$\sqrt{n} (X_n - \mu) \to ...
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1answer
32 views

Rate of expected value of $\mathcal{O}_p$

This is certainly very basic but what is the rate of the expected value of a random variable that is bounded in probability. For example, let $X_n = \mathcal{O}_p (a_n)$ is it true that $\mathbb{E} ...
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6 views

Determining Convergence of a Data Set using Moving Variance

I'm doing some research with Matlab, and a lot of my work involves statistical analysis, of which I'm not super familiar with (so I apologize in advance if I'm not using proper terminology). For this ...
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26 views

Converge in Probability of random variables

I don't know if I understood the (Convergence in probability of random variables) formula, why $ | Xn -X | $ should be $$ \geq \varepsilon $$ For example if $\varepsilon $=5( a random number), and ...
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1answer
216 views

root-n consistent estimator, but root-n doesn't converge?

I've heard the term "root-n" consistent estimator' used many times. From the resources I've been instructed by, I thought that a "root-n" consistent estimator meant that: the estimator converges on ...
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1answer
39 views

Confidence interval for MLE estimator?

I have an MLE estimator which is asymptotically normally distributed with mean $\beta$ and variance $\beta^2/n$. How do I get an approximate confidence interval for this estimator? I know usually two ...
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76 views

How to check the convergence in the collapsed Gibbs sampling of LDA? [closed]

I am trying to implement the LDA model fit by collapsed Gibbs sampling by myself. I have go through this article. And there is a clear pseudo code (section 5.5), ...
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2answers
205 views

How does Slutsky's theorem extends when two random variables converge to two constants?

The 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 element $X$ and $Y_n$ converges in ...
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224 views

Yet another central limit theorem question

Let $\{X_n:n\ge1\}$ be a sequence of independent Bernoulli random variables with $$P\{X_k=1\}=1-P\{X_k=0\}=\frac{1}{k}.$$ Set $$S_n=\sum^{n}_{k=1}\left(X_k-\frac{1}{k}\right), \ ...
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37 views

Poisson Convergence Example

For any $n\ge1,$ let $X_{n1},...,X_{nn}$ be independent with $$P\{X_{nk}=5\}=1-P\{X_{nk}=0\}=p_{nk}, \ k=1,...,n$$. Assume that $$\lim_{n\rightarrow \infty} \max_{1\le k \le n} p_{nk}=0$$ and ...
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20 views

Expectation of discrete random variable

Give a sequence of random variables $x_1,..,x_n$ with $x_n$ having a density of: $$f_N(x) = \begin{cases} \frac{2N-1}{3N};x=1\\ 1/3;x=1+\frac{1}{N+1} \\ \frac{1}{3N};x=2\end{cases}$$ What would be ...
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Python statsmodels ARIMA ValueErrors

I am using the statsmodels package in python to generate a set of ARIMA models for a series of log returns multiplied by 1000. I am iterating through possible models (p, d, q) starting from (1, 0, 0) ...
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58 views

Convergence of Monte Carlo - Why square root n for rate?

Let $\theta_n = \frac{1}{n} \sum_{i=1}^n X_i$ be the Monte Carlo estimator for $E(X)$. Letting $\sigma^2 = \operatorname{Var}(X)$, by the CLT, $$ \sqrt{n}(\theta_n - E(X)) \xrightarrow d ...
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44 views

steps to try and diagnose why fitting a logistic regression fails

When fitting a logistic regression (or any regression fit by maximum likelihood) fails, we sometimes get an informative message. For example, lrm in the R rms package provides messages such as: ...
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43 views

Why is it that a larger 'k' value fails to converge but a smaller 'k' converges?

I'm doing clustering via GMM, which is initialized first by k-means. I am using a data matrix that cannot be classified as small by any standards, they are usually of the size ...
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56 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 ...
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93 views

Using the central limit theorem

Use the central limit theorem to show that for $x>0$, $$\lim_{n \rightarrow \infty} \frac{1}{3^n} \sum_{k:|3k-2n| \leq \sqrt{2n}x} \binom{n}{k} 2^k = \int^{x}_{-x} ...
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65 views

Why does $\frac{\sum X_i^2}{n} \rightarrow \sigma^2+\mu^2$?

I cannot find a reference to the proof of this, but I found $$\frac{\sum X_i^2}{n} \rightarrow \sigma^2+\mu^2$$ From: http://math.stackexchange.com/a/1416911/248602 So why does it converge to ...
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41 views

strong law of large numbers for random vectors

It is well-known that if $X[n]$ is a sequence of i.i.d. random variables, each with mean $\mu_X$, then as $n\rightarrow \infty$, we have $\mu_X[n]\triangleq \frac{1}{n}\sum_{k=1}^n{X[k]}\rightarrow ...
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63 views

Determine distribution of a limit of random variables

Suppose a box contains a blue ball and green ball. After every hour a ball is chosen from the box randomly and then is put back with another ball of the same colour. After $h$ hours, there are ...
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33 views

Convergence in probability in high-dimensional settings

I am trying to prove the following. $$ (\hat{\theta} - \theta)^T c \rightarrow 0 \;\;\; \text{ (in probability)}$$ where $\theta \in R^p$, $c$ is a vector of constants such that $\sup_i (|c_i|) ...
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1answer
37 views

Logistic regression with r and stata

I ran the same Logistic regression with R and STATA. The regressors include many dummy variables. In R, the code I used is ...
3
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1answer
52 views

When does the marginal MLE converge to the complete data MLE?

What I mean by the title is suppose we have a distribution $p(x,z\;|\;\theta)$, where the $x$ are observed and each $x_i$ depends on a hidden $z_i$. Then the marginal MLE is given by ...
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34 views

Why does the basic gradient descent not converge for this example?

I have a toy example for a linear regression of the form $$y=\beta_0 + \beta_1x_1 + \beta_2x_2$$ The data is: ...
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Distibution of the pdf as a random variable when evaluated at samples generated from itself

Let $x_i \sim p$ for some probability density function $p$ with respect to Lebesgue measure on $\mathbb{R}$. Then each of $p(x_i)$ is a random variable taking values in some interval $[0,c]$ from some ...
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15 views

Instantaneous drift of a stochastic process

(Posted also on math.stackexchange) Let $\mu_t$ and $\sigma_t$ be strictly positive bounded predictable processes and $W_t$ a Wiener process. Consider for $\Delta>0$ $$ X_{\Delta} = ...
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28 views

Stopping Criteria for ADMM algorithm

In Page 19 of this paper admm_distri_stats, the authors provide one reasonable stopping criteria for ADMM algorithm in (3.12). Can anyone tell me if there is any principle to find the reasonable ...
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If all components of a hierarchical model have not converged, can we say that any parameters have truly converged?

I'm working with a hierarchical regression model of the following form similar to that presented in Peter D. Hoff's book, A First Course in Bayesian Statistical Methods: $\boldsymbol{Y}_j \sim ...
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24 views

How to test whether a time series of measurements have converged to an equilibrium

I have a time-series of data that looks like this (as a couple of examples): This mean energy is the mean over a number of Monte Carlo test particles. The number of particles vs. time is not ...
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11 views

canonical benchmark on convergence measures for neural networks

This is in reference to an answer to a previous question (here), and to a related question (here). I know there are a truck-load of data sets out there (link). I know there is very wide variety of ...
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How to decide about the number of looks (window size for ensemble averaging) in SAR images?

This question has frustrated me for a while. In order to find an answer I sent an email to prof. Yamaguchi, the author of the paper Four-Component Scattering Power Decomposition With Rotation of ...