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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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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82 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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49 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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53 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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79 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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193 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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21 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
29 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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199 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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37 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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74 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
197 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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222 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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1answer
36 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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19 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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53 views

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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48 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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42 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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54 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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63 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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37 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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57 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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31 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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34 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 ...
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51 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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33 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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14 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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22 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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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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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 ...
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53 views

Inverse gamma convergence in probability

I am trying to prove the consistency of MLE for a beta distribution. The problem now reduces to the following: Assume $Y=\frac n X$ and $X$ ~ Gamma(n,$\frac 1 \theta$), prove that $Y$ converges to ...
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33 views

SPSS: GLMM and(adjusted) odds ratio

I am performing a retrospective study and the relative statistic analysis. I am studying the the risk factors for the occurrence of complications during medical procedures. I have 50 subjects ...
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12 views

Improving convergence of simulated annealing

I have a complex curve that I need to approximate with a parametric model (5 parameters). The parametric model itself is highly non-linear, and small changes in parameters can lead to big changes in ...
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27 views

Convergence of a sequence of Binomial variable with changing probability

Consider a $t\in(0,1)$. Consider, for $\Delta>0$ the random variable $X_t^{(\Delta)}$ defined as $$ \mathbb{P}[X_t^{(\Delta)}=1]=\left(1-\lambda\,\Delta\right)^{\left\lfloor ...
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76 views

convergence of Cauchy distribution

It is known that the Large Number Theorem does not apply to Cauchy distribution since it does not have an expectation value. That said, $S_n / n$ does not converge in any sense (almost sure, in ...
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24 views

Asymptotic normality for nonsmooth objective functions

Assume that $f ({\bf x}; \theta): \mathbb{R}^p \times \Theta \to \mathbb{R}$, where ${\bf x}$ is the vector of inputs (with some distribution) and $\theta$ is the vector of parameters. Also, assume ...
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Continuous mapping theorem for convergence in probability

I have seen the continuous mapping theorem (CMT) used to justify the convergence in probability of the difference of two sequences of random variables when it is known that each sequence converges in ...
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306 views

Statistical test to verify when two similar time series start to diverge

As from title I would like to know if exist a statistical test that can help me to identify a significant divergence between two similar time series. Specifically, looking the figure below, I would ...
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70 views

Convergence in probability question

Being a novice to this topic, I am not being able to properly write down a step by step solution to this problem. For each integer $n$, let $X_n$ be a non negative random variable with finite ...
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40 views

Is there a central limit theorem for i.n.i.d. variables when normalised by inconsistent variance estimate?

I am wondering whether there exists a central limit theorem for the following situation. Consider the sum of normally distributed variables $\epsilon_i$ with unequal variances according to ...
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27 views

A convergence problem

Let Xn be a sequence of random variable that converges in distribution to a random variable X, Let Yn be a sequences of random variables with the property that for any finite number c, ...
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118 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 ...
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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 ...