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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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
32 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
43 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 ...
2
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1answer
28 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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2answers
130 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
49 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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15 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
77 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
31 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) = 1āˆ’\theta $ for $\theta \in (0, 1)$ an unknown parameter. Suppose an i.i.d. sample of size ...
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2answers
42 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
30 views

Textbook GMM/Convergence Question

Consider a binary random variable X that satisfies: $Pr(X = 0) = \theta \ \ \ $ and $Pr(X = 1) = 1āˆ’\theta $ for $\theta \in (0, 1)$ an unknown parameter. Suppose an i.i.d. sample of size $n$ ...
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3answers
143 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
33 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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45 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
46 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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40 views

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

Iā€™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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0answers
20 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 ...
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1answer
73 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
66 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
22 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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31 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
68 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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20 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
40 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
49 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
27 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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14 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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21 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 ...
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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
54 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
43 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?
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1answer
87 views

Show that the sample covariance converges in probability to the $Cov(X,Y)$

Suppose we are given $[(Y_i,X_i)]^{n}_{i=1}$ which is a random sample from the joint distribution of $(Y,X). Show that the sample covariance converges in probability to the $Cov(X,Y)$ My thought ...
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5answers
501 views

Intuitive explanation of convergence in distribution and convergence in probability

I am quite unsure of the intuitive difference between a random variable converging in probability versus a random variable converging in distribution. I've read numerous definitions and mathematical ...
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1answer
28 views

What to do about very unstable mixed-effects models

I'm working on some poisson mixed effects models for an interrupted time series analysis, and I'm running into two frequent errors. The first I've posted on Stack Overflow, as it appears to be purely ...
3
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1answer
40 views

What is the optimal bandwidth in this problem

Suppose that I have some estimator $\hat f$ for the parameter $f$. I obtain the following convergence rate $$E\|\hat f - f\|^2 = O\left(\frac{1}{nh} + \frac{1}{nh^2} + \frac{1}{\sqrt{n}h} + ...
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2answers
424 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 ...
3
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0answers
94 views

Does $\sqrt{n}(Y_n-1)$ converge in distribution?

Let $n\in \mathbb{N}$ and consider, for each $n$, independent real valued stochastic variables $Z_{1n}\ldots Z_{nn}$, such that $$ P(Z_{nk}=n)=1-P(Z_{nk}=0)=\frac{1}{n} $$ for $k=1,\ldots,n$. Thus ...
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25 views

Convergence of distributions implies convolution (with itself) converge?

If 2 distributions converge in the limit, then do their convolutions also converge? e.g. I can show that for the Geometric distribution with random variable $T$, the scaled version with parameter ...
3
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1answer
156 views

Convergence from Gamma to Normal Distribution

I came across this problem: Problem If I have $X_1, X_2, ..., X_n$ $n$ iid random variables which pdf is $$ f_X(x) = \begin{cases} \dfrac{x^{\mu-1} e^{-x}}{\Gamma{(\mu)}} &0<x<\infty, ...
3
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1answer
50 views

Converging exponentially in probability implies convergence with probability one ?

When I read the paper On the Strong Universal Consistency of Nearest Neighbor Regression Function Estimate, the theorem 1 in it states something like If for every $\epsilon > 0$ there exists ...
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95 views

My GARCH model does not converge in Stata

I'm trying to model impact of oil supply shocks on stock prices using GARCH methodology. I have a sample of countries, some for which the GARCH model converges and gives reasonable estimates. For some ...
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66 views

Correctness of a proof for Hodges' estimator

We know the following is Hodges' estimator: $$ \delta_n = \begin{cases} \bar{X}_n & |X_n| \geq n^{-1/4} \\ a\bar{X}_n & |X_n| < n^{-1/4} \\ \end{cases} $$ where $X_1, ..., X_n \sim ...
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1answer
40 views

Quadratic mean convergence of a biased coin using conditional expectation

I'm a master's degree student and after a lot of research and some days trying I still can't get the answer for a question proposed by my Statistics professor. He asks to toss a coin with a random ...
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97 views

GLIMMIX convergence problems with ordinal model

I am analyzing a longitudinal data set about hearing loss which has 226 subjects with repeated measures each one (with a maximum of 12 observations per subject) over a follow up time of 22.2 years. I ...
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2answers
137 views

Pointwise convergence in probablity of lasso

In the Knight and Fu's paper, in Equation 6 authors consider the pointwise convergence in probability as $$\underset{\phi \in K}{\operatorname{sup}} | Z_n(\phi)-Z(\phi)-\sigma^2| \longrightarrow_p ...
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37 views

Convergence Time of the EM Algorithm Depending on the Inital Parameter Values

I try to get an intuitive understanding of the convergence properties of the EM-Algorithm. I wrote a code that does the following experiment. We are given three coins: $H$, $A$ and $B$; with ...
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1answer
122 views

Convergence of likelihood implying convergence of marginal likelihood?

I will ask my question through a toy motivating example. It is well known that a Poisson process is the continuous time analog to a Bernoulli process (for example: ...
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1answer
18 views

Max fraction of censored values for Cox regression?

I'm not sure whether there's an exact answer to this, but I'm wondering in the simplest case of doing Cox Regression with censoring on one variable, if we have N measurement values, what number (k) of ...
3
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1answer
43 views

Convergence in probability for two statistics of Laplace random variables

Suppose $X_i$ are iid random variables. $X_i \sim \mathrm{Laplace}(\lambda)$. Also define: $$ U_n = \frac{1}{n-2}\sum_{i=1}^n{|X_i|} $$ $$ V_n = \sqrt{\frac{1}{n-1} \sum_{i=1}^n{X_i^2} } $$ Given ...