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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Linear Regression and Almost Sure Convergence

Consider a linear regression model, wherein: $$ y_{i}=x_{i}\beta+\epsilon_{i} $$ where notation is standard and $x$ is a scalar. Let us further impose the following restriction: $$ \epsilon_{i}|x_{i}\...
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Confused about the Convergence Theorem for a Generalized MDP

Sorry for the wall of text below. It will seem like forever before my question comes up, but I think the context is much needed to avoid confusion. According to "A Generalized Reinforcement Learning ...
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17 views

got msg “convergence warning in initialization\n” running svmpath() [closed]

I got msg "convergence warning in initialization\n" running svmpath(). Data consist of 15 factors, transformed by "model.matrix" into 84 constrasts; 4452 instances. ...
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20 views

Multiple chains of MCMC don't agree [closed]

I am running 4 chains in parallel and it turns out that the likelihood of all these chains stay quite stationary at the end of the chains, but they differ from each other a lot. I am not sure if I ...
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19 views

Is linear regression appropriate for measuring convergence/stabilization?

Background I'm running a simulation that produces a number of outputs on every iteration. The simulation runs for a good half a million iterations, and my goal is to check if it stabilizes or I ...
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60 views

Inverting Laplace transform

From Williams' Probability w/ Martingales: Are we allowed to switch derivative and integral as follows: $$\frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x} f(x) = \int_{0}^{\...
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1answer
42 views

What's the convergence rate in the context of convergence in probability?

A sequence $z_n$ with $\lim_n z_n = z$ is said to have $Q$-linear convergence if a constant $r\in (0,1)$ exists such that $\displaystyle |z_{n+1} - z| \leq r \, |z_n - z|$, where $r$ is called the ...
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15 views

What do we know about the rate of convergence of the mean of RVs with infinite variance?

What, if anything, do we know about the rate of convergence of of the mean of identically distributed, independent or stationary random variables drawn from a distribution with a finite mean and ...
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1answer
49 views

Problem on convergence of sequence of random variables

Given any sequence of random variables $\{ X_n \}$; how do I show that there exists a sequence of real numbers $\{ \alpha_n \}$, such that $\{ \alpha_n X_n \}$ converges in probability to 0 ?
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1answer
35 views

Continuous uniform random variables convergence question

Let $X_1, X_2, \ldots$ be independent $U(0,2)$ random variables and let $$Y_n = \prod_{i=1}^n \, X_i \;.$$ How do I prove or disprove that that $Y_n$ converges to $0$ almost surely ?
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18 views

HMM Training: Testing convergence

What is the best way to test for convergence while training an HMM? I understand that we need to iterate till the change in parameters ( transition matrix, emission matrix ) is less than the threshold....
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15 views

Limiting distribution with two additive terms converging in distribution

I have a question related to the asymptotic distribution of an estimator $\hat{\beta}$ that tries to estimate the true parameter vector $\beta$ and can be written as $\sqrt{N}(\hat{\beta}-\beta)=\...
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1answer
66 views

Convergence to zero of a sum involving the median

Can anyone help me with this? Let $\{X_i:i=1,...,n\}$ be a sequence of i.i.d random variables with a continuous density function. If $m$ is the true median and $\hat m _n$ is the sample median, is it ...
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38 views

Sufficient conditions for variance convergence in CLT

More generally, if $\{X_n\}_{n\in\mathbb{N}}, X$ are real random variables with finite variance such that $X_n\xrightarrow{d}X$, what are some sufficient conditions to assure that $\operatorname{Var}(...
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28 views

When 2% of the Bayesian Model have not converged?

I have model with 20000 latent parameters, set up in a Gibb's sampler. 98% of the parameters and sometimes 99.5% of the parameters satisfy the Geweke convergence statistic, have low autocorrelation ...
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35 views

Non-convergence in nonlinear regression

I've been studying a manuscript from the chemistry literature (not mine) that resorted to a trick to obtain convergence of a non-linear model fitted to experimental data. They wanted to estimate a ...
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17 views

Is supremum continuous on D[0,1]? Needed for continuous mapping theorem

I am currently in the following situation: I have a sequence of processes which converges weakly to a limit process in D[0,1]. I want to show that the supremum of the processes converges as well to ...
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29 views

Why exactly is it that $\sup E[M_n^2] < \infty \iff E[A_{\infty}] < \infty$

Probability with Martingales: About $(c)$ My understanding is that $(c)$ is equivalent to: $$\sup E[M_n^2] < \infty \iff E[A_{\infty}] < \infty$$ Since $E[M_n^2] = E[A_n]$, we have ...
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1answer
27 views

Ideal Inertia for k-mean clustering convergence

Is there an ideal "inertia" for K-mean convergence. For example I'm trying to cluster to 64 clusters using sci-kit. the output is ...
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1answer
24 views

Self study - Determining function codomain to study convergence

I'm having a serious problem in and old exam paper, specifically a question on convergence of random variables. Let $X$ and $Y$ be two i.i.d exponential random variables with parameter $\lambda$, so ...
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63 views

Show estimate converges to percentile through order statistics

Let $X_1, X_2, \ldots, X_{3n}$ be a sequence of iid random variables sampled from an alpha stable distribution, with parameters $\alpha = 1.5, \; \beta = 0, \; c = 1.0, \; \mu = 1.0$. Now consider ...
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25 views

What is a hard data set for support vector machines?

Suppose that: There are only numeric features in the data set. There is a binary target variable (since a general nominal case is resolved by 1vs1 or 1vsTheOthers approaches). I have noticed that ...
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22 views

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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1answer
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 $\frac{1}{2n}$} \end{...
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16 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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14 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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20 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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1answer
25 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
30 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 $ M_{x_n}(t)=(1-2t)...
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51 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} u^...
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1answer
64 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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1answer
43 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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57 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
111 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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20 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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181 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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53 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
56 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
91 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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2answers
208 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 distribution?...
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26 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 \mathcal{...
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1answer
35 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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9 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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29 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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294 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
40 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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103 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), ...
2
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2answers
211 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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3answers
227 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), \ B_n^2=\sum^{n}_{...