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 Bernoulli sampling procedure

Let $X$ be a variable which has density $f_{X}$ and mean $E[X]=\mu$ Define $B$, a Bernoulli variable, which has a probability $P(b_{i}=1)=p_{i}=\frac{1}{N^\gamma}$ Consider the following ...
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24 views

convergence rate of Pearson correlation matrix

I posted recently a related question about the convergence rate of a Pearson correlation coefficient, here. Now, I am interested in the matrix version. Let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be ...
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2answers
42 views

Convergence of moments of binomial to Poisson

As is well known, the $\mathsf{Binomial}(n,p)$ distribution converges to the $\mathsf{Poisson}(a)$ distribution as $n\rightarrow \infty$, $p\rightarrow 0$ with $np=a$. I'm pretty sure that the ...
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19 views

Theoretical properties of Gaussian Process Emulator

I am studying Guassian Prcess Emulator (GPE) to approximate computationally expensive computer models. Basically, we suppose the computer model, or simulator, is denoted by $f(x)$, where $x$ is the ...
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1answer
33 views

Does convergence matter in finding the best fit?

I am trying to fit my model using garchFit function in the fGarch package, however I keep on getting this error: ...
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1answer
23 views

How to show the a sequence converges to 0 almost surely

There are two nonnegative sequences $a_n$ and $b_n$. We know that $a_nb_n$ is summable a.s., i.e., $\sum_{n=1}^\infty a_nb_n < +\infty$ a.s.. $a_n$ is not summable a.s., i.e., $\sum_{n=1}^\infty ...
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1answer
139 views

MLE and Bayesian methods

I saw in some lecture the fact that as the number of data points N goes to infinity, the prediction of the Bayesian method goes to the prediction of the MLE. Can someone explain what exactly this ...
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16 views

Detecting Gibbs Sampler convergence with Raftery and Lewis Diagnostic

Hi! I'm trying to understand and implement the Raftery and Lewis Diagnostic for detecting the number of iterations required for a gibbs sampler but cant seem to understand the formula. ...
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1answer
37 views

How many AdaBoost iterations?

In one R package, ada, the main AdaBoost fitting function (also called ada) takes an argument specifying the number of boosting ...
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1answer
37 views

What does 'iteration limit reached without convergence' mean in SAS Proc PLS

I receive the warning 'iteration limit reached without convergence' when using PROC PLS in SAS. What does this warning mean? I have 1,540 observations, 900 dependent variables and 600 independent ...
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6 views

Polyak-Juditsky and Robbins-Monro Stochastic Estimators

Background/Motivation: I just read on Wikipedia that the Robbins-Monro estimator in its original form, while allegedly capable of O(1/n) convergence tends to be highly dependent on stepsize, and ...
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31 views

Convergence in distribution

Say I have n boxes and m balls. The balls have distribution of colours given by F. The size of the boxes varies with distribution G. Each box has t balls and the number of balls is given by the ...
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34 views

Convergence in distribution of the sum of two RVs

Let $X_n$, $Y_n$ and $Z_n$ be RVs such that $Z_n = X_n + Y_n$. Assume that $X_n$, $Y_n$ have joint distribution $F_n\left(x, y\right)$ and $X_n$ has a distribution function $F_n\left(x\right)$. Let ...
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32 views

Does something justify using bglmer to correct the warnings I got with the glmer

I conducted a risk factor analysis for which I got some warning messages. Data: bf= if the piglet got the disease=1 if the piglet didn't=0 y= year (categorical data) SOW= random effect of the ...
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1answer
43 views

OLS estimation for Nonlinear model

Consider the following model which may be nonlinear: $Y_{t} = f (X_{t}, \beta_{0}) + \mu_{t}, \hspace{0.2cm} t=1, ..., T$ If we assume that: $\mu_{t}$ i.i.d with mean = $0$ and ...
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2answers
60 views

Understanding Big/Little $O_p$/$o_p$ Notation for Estimators

I am reading a Text about Single Index Models (SIM), where a SIM is defined as $E[Y|X=x] = G(X' \beta)$, with $G$ and $\beta$ unknown. After proposing an estimator for the function $G$, the ...
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0answers
28 views

Planned missing data design not converging

I recently ran an experimental study with a planned missing design. Participants were randomly assigned to one of four groups that dictated which portion of a 26-item unidimensional measure they ...
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21 views

Imposing singular value constraints on recurrent neural networks

Apologies for beginning with what are surely very old results, to make clearer my thinking. First, note that any multi-layer perceptron (MLP) may be represented as a recurrent neural network (RNN). ...
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165 views

A continuous function of a sequence of random vectors converges in probability to the function of the limit

Proposition: If $\{ X_n \}$ is a sequence of k-dimensional random vectors s.t. $X_n \overset{p}{\to} X$ and if $g: R^k \rightarrow R^m$ is a continuous mapping, then $g(X_n) \overset{p}{\to} g(X)$. ...
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1answer
40 views

Proving the sum rule of probability limits

I'm trying to show that if $X_n$ converges in probability to 0 and $Y_n$ converges in probability to 0, then $X_n+Y_n$ converges in probability to $0$, ie the sum rule for probability limits. What ...
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1answer
86 views

Convergence of EM for Mixture of Gaussians

Is the Mixture of Gaussians model (an example of latent class analysis) gauranteed to converge on a viable solution even on Unimodal data using the Expectation Maximization algorithm to estimate the ...
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1answer
37 views

Central limit theorem on a linear combination

I am looking for the name and the formulation of a CLT variant that states that a linear combination of random variables with the same mean and standard deviation will converge under a specific ...
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143 views

Convergence of a product of random variables

Let $X_1, X_2, \dots $ be a sequence of I.I.D. random variables with pdf $f(x) = \frac{8x}{9}, 0 < x < 1.5$. What does the product $\prod_1^nX_i$ converge to in the almost sure sense? Shouldn't ...
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1answer
51 views

SLLN applies to last $r_n$ variables?

Let $X_n$ be a sequence of IID random variables with finite mean and first moment, let $S_n = \sum_1^n X_n $ then is it true that \begin{equation} \frac{S_n - S_{n-r_n}}{r_n} \end{equation} ...
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44 views

How to interpret the results of geweke.diag() function present in Coda package of R?

I am using geweke.diag() to check the convergence of an MCMC chain, I am using following R-Code for the purpose ...
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1answer
8 views

Convergence rate when a heavierside function is involved

Let $\bf \alpha \in {R}^p$ be a parameter, $\bf \alpha_0$ is the true and $\tilde{\bf \alpha}$ is its estimator such that $\|\tilde{\bf \alpha} - {\bf \alpha}_0\|_2 = O_p(N^{-r})$, where $N$ is the ...
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1answer
109 views

Slutsky's Theorem to show convergence to Standard Normal Distribution

We are given $W_n = \frac{\bar{X}-\lambda}{\sqrt{\bar{X}/{n}}}$ and need to show it converges to a standard normal distribution. EDIT: The square root in my original post did not extended over the ...
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1answer
41 views

Convergence of standardized means of a Bernoulli variable / CLT

The Question Consider a binary random variable X that satisfies: $Pr(X = 0) = \theta \ \ \ $ and $Pr(X = 1) = 1−\theta $ for $\theta \in (0, 1)$ an unknown parameter. Suppose an i.i.d. sample of size ...
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2answers
43 views

Check my proof regarding convergence in probability

I got a bit confused during the end of this proof so I am asking for a check. Take $$Y(n) = \begin{cases} 1 &\mbox{with probability} \ 1 -p_n \\ n & \mbox{with probability} \ p_n \end{cases} ...
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1answer
31 views

Textbook GMM/Convergence Question

Consider a binary random variable X that satisfies: $Pr(X = 0) = \theta \ \ \ $ and $Pr(X = 1) = 1−\theta $ for $\theta \in (0, 1)$ an unknown parameter. Suppose an i.i.d. sample of size $n$ ...
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148 views

Regarding convergence in probability

Let $\{X_n\}_{n\geq 1}$ be a sequence of random variables s.t $X_n \to a$ in probability, where $a>0$ is a fixed constant. I'm trying to show the following: $$\sqrt{X_n} \to \sqrt{a}$$ and ...
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1answer
35 views

Convergence in Mean Square

Let $E[y_t|y_{t-j},y_{t-j-1},\cdots] \xrightarrow[m.s.]{}0$ as $j \rightarrow \infty$.Is it necessarily true that $E[y_t] = 0$? My Attempt: \begin{align*} E[y_t|y_{t-j},y_{t-j-1},\cdots] ...
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53 views

$X_n$ $\,{\buildrel a.s. \over \rightarrow}\,$ $X$, then $(\prod_{i=1}^{n}X_i)^{1/n}$ $\,{\buildrel a.s. \over \rightarrow}\,$ $X$?

Prove or provide a counterexample: If $X_n$ $\,{\buildrel a.s. \over \rightarrow}\,$ $X$, then $(\prod_{i=1}^{n}X_i)^{1/n}$ $\,{\buildrel a.s. \over \rightarrow}\,$ $X$ My attempt: FALSE: Suppose ...
3
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1answer
57 views

Gibbs Sampler - Sample mean convergence

To simulate from the posterior distribution $p(\theta|Y)$ where $\theta = (\mu,\lambda_1,\lambda_2)$, I run a Gibbs sampler to draw approximately random values from $p(\theta|Y)$. This Gibbs sampler ...
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57 views

Can complete separation between a continuous predictor and a random effect cause failure to converge in a logit GLMM?

I’m running a logit mixed-effects model on binary data with a 2x2 within-subjects design, with subjects and items as crossed random effects, and the two independent variables deviation-contrast coded. ...
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39 views

How to measure convergence of data?

I am running a word-game simulations that assign values to words. In short, a computer is self-playing games of Scrabble and each move is recorded as (word, point value of the word). One value usually ...
5
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1answer
75 views

Prove $X_i$ in Span of $ X_k, k \neq i$

Let $(\Omega,\mathcal F, P)$ be a probability space and let $L_1$ be the collection of all integrable (finite expectation) real-valued random variables defined on this space. Assume that for $X_k\in ...
5
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1answer
82 views

Almost sure convergence does not imply complete convergence

We say $X_1, X_2, \ldots$ converge completely to $X$ if for every $\epsilon>0$ $\sum_{n=1}^\infty \text{P}\left(|X_n-X|>\epsilon\right) <\infty$. With Borel Cantelli's lemma is straight ...
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1answer
24 views

Convergence of EM in Mixture Models w.r.t unlikely events $(f(\cdot)=0)$ in either distribution

To maximize the likelihood of a mixture model with unobserved latent variables, the Expectation Maximization is conventionally applied. Assuming we have data $x_1,\dots,x_n$ from a fixed number of ...
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38 views

Relationship between convergence in probability and convergence in distribution

I am a little confused about whether something converges in probability and/or in distribution? If we had a sequence of random nummbers $\{ X_n\}$ with $EX = 0$ and $EX^2 = 1$, I know from the ...
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1answer
69 views

Going from derived estimators to their implementation in software

Estimation and Inference in Econometrics by Davidson and MacKinnon (1993 edition, the older one) on page 552, ch 16.3 'Covariance Matrix Estimation' states: "Consequently, the matrix \begin{eqnarray} ...
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24 views

Uniform Convergence of Moment under Empirical distribution

Let $X$ be standard Gaussian random variable with cdf $F(x)$. Let $\{X_i\}_{i=1}^n$ be a sequence of i.i.d. standard Gaussian random variables. And let $F_n(x)=\frac{1}{n}\sum_{i=1}^n1_{\{X_i\leq ...
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2answers
50 views

Suspiciously high Multivariate PSRF from gelman.diag()

I am using "Multivariate PSRF" statistics from gelman.diag() function to analyze my MCMC chains. Now I analyzed convergence 471 variables (parameters for each ...
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1answer
58 views

Divergent sequence of random variables with expected value zero

I am trying to think of a sequence of random variables $X_n$ such that $X_n\rightarrow\infty$ as $n\rightarrow \infty$ but $E(X_n)\rightarrow0$. Can you please show me some examples?
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1answer
32 views

Convergence of Sum of Sums of random variables : trivial?

I have a sequence of i.i.d random varaibles $X_1, X_2, ...$ with finite mean $\mu$ and finite variance. I also have another sequence of i.i.d random varaibles $Y_1, Y_2, ...$ with the same finite mean ...
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16 views

Likelihood convexification

I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form: $$\begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log ...
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28 views

pLSA using EM not converging

I have found a lot of questions related to EM but nothing specific to my question. I am using the EM algorithm to fit the pLSA model. As far as I can tell (multiple rounds of checking the code) I can ...
2
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62 views

MCMC convergence: why Heidelberg's test says normal samples are non-stationary?

I am learning about and playing with Heidelberg's convergence test to automatically stop a MCMC sampling. I would have said that if I sample, for instance, from a normal distribution, the test ...
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1answer
67 views

Convergence in distribution with empirical distribution function (EDF)

I'm struggling with the following question: Let $F_n(x)$ denote the EDF of a random sample. Show that $\sqrt{n}(F_n(x)-F(x))\xrightarrow[]{d}N(0,F(x)(1-F(x)))$. I think that the course of ...
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2answers
46 views

What is meant by the term “convergence” in Restricted Boltzmann Machine?

I have come across the term "convergence" in training RBM. Can someone give a brief definition / explanation of it?