Questions tagged [convergence]

Convergence generally means that a sequence of a certain sample quantity approaches a constant as the sample size tends to infinity. Convergence is also a property of an iterative algorithm to stabilize on some aim value.

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Intuition of the convergence of sample ACF

One of the problems in Brockwell and Davis book about time series is to show that 1) if \begin{equation} x_t = a + b t \end{equation} then the sample autocorrelation ($\hat{\rho}(h)$) converges to ...
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How to prove absolute summabilities implies the absolute summability of the product series?

In SHUMWAY 2017 Time Series Analysis and Its Applications with R examples 4E, page 486, it states: $\Sigma_{j=-\infty}^{\infty} |a_j| < \infty$ and $\Sigma_{j=-\infty}^{\infty} |b_j| < \infty$ ...
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Why does absolutely-summable weights ensures a linear series itself summable (convergent)? Some questions on def'n of Linear Series

A "linear series" $y_t$ is the linear combination $$y_t - \mu = \sum_{i=-\infty}^{\infty}\psi_iL^i\nu_t = \sum_{i=-\infty}^{\infty}\psi_i\nu_{t-i}=S(L)\nu_t $$ of weighted (by $\psi_i$ weights) lags ...
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Convergence rate of the maximum of Weibull random variables to a Gumbel distribution

Given a sequence of iid samples $X_1, \dots, X_n,$ where each $X_i$ comes from a Weibull distribution with shape parameter $k$ and scale parameter $\lambda$. Then it is a well-known result that the ...
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Convergence of empirical quantiles to theoretical quantiles - mixed type distribution

It is well known that under certain (not too restrictive) conditions empirical quantiles of a distribution converge to the corresponding theoretical quantiles in probability as the sample size $n \to \...
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If $y_t$ is a time series with autocovariance $\gamma$, does $\gamma$ necessarily have to be absolutely-summable?

If $y_t$ is a time series with autocovariance $\gamma$, does $\gamma$ necessarily have to be absolutely-summable; i.e., ${\sum_{i=\infty}^\infty |\gamma (i)}|<\infty$? If not, what could be the ...
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pymc3: finding lowest misfit using MCMC [closed]

I have made a function that calculates the misfit(squared difference) between an observed signal at a target station and reconstructed signal from 10 arbitrary stations chosen from a dataset. There ...
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Beneficial dimension for 2nd order modelling in SGD optimization?

There are currently mostly used first order methods in SGD optimizers, second order are often seen too costly as e.g. full Hessian has size $D^2$ in dimension $D$. But we don't need full Hessian - ...
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What is the virtue of loading absolutely-summability in the definition of causality of ARMA model?

An ARMA series $y_t$ is causal function of $\nu_t$ if there exists constants $\psi_j$ such that $\sum_{j=0}^{\infty} |\psi_j|<\infty$ and $y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}<\infty$ for ...
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Convergence warning in LME4 despite p~=0.05 via ANCOVA? [migrated]

I’m having trouble with convergence warnings using lme4. I'm collecting time-series data. The outcome measure ("TCV") is derived from electromyography. I run repeated measures tests on the same ...
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Monte Carlo simulation percentile convergence

Suppose I have a normal distribution random variable $X$ with mean $\mu$ and variance $\sigma^2$. I can easily calculate the 99.7 percentile, which $\approx\mu+3\sigma$. In the Monte Carlo simulation ...
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Is there any statistical test to confirm if a dependent variable converges to a value as the independent variable approaches infinity?

I have a very large table with 40,000 elements, and the dependent variable appears to approach a value as the independent variable gets larger. Is there any test I can perform to confirm if the ...
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Convergence in law of the remedian

I try to understand the theroem 2 of this article about the remedian (https://pdfs.semanticscholar.org/3d64/5e60691838bf4699e79458d96930ba7bf24e.pdf) I will try to phrase it more general so that you ...
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Is this series divergent?

I have $$G_N = \sum_{i=1}^{N} \left\{ \frac{1-\pi_i}{\pi_i} + \frac{1-\pi_i}{T\pi^2_i}\right\} (y_i-\theta)^2=\sum_{i=1}^{N} V_i$$ where $2\le T\le 10$, $0\le \pi_i\le1$ and suppose $y_i\sim N(\theta,...
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Low effective sample size but good R-hat is this a problem?

I am using Stan (Hamiltonian Monte-Carlo) to run a highly paramaterized model. One of the parameters in particular has a very low effective sample size (n_eff < .10*number of retained draws), but ...
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MCMC with slowly varying Log-Likelihood

I am using MCMC (Metropolis-Hastings) to simulate values of $\theta$: I have a Log-likelihood (using 10 inputs $x_i$) $$L=-\frac{n}{2}\ln(2\pi)-\frac{1}{2}\sum_{i=1}^n(x_i-\theta)^2$$ The variation ...
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Why is a Gelman-Rubin diagnostic of < 1.1 considered acceptable?

In multiple sources a Gelman-Rubin MCMC convergence diagnostic of less than 1.1 is considered evidence that chains have converged. For example in this thread: https://stackoverflow.com/questions/...
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Matrix inversion and $\mathcal{O}_p$ notation: $(\mathbf{A} + \mathcal{O}_p(f(n)))^{-1} = \mathbf{A}^{-1} + \mathcal{O}_p(??)$

Suppose that $\mathbf{A}$ is square invertible matrix and that $\hat{\mathbf{A}}$ is an estimator of $\mathbf{A}$ based on a sample of size $n$ such that: $\hat{\mathbf{A}}$ is invertible, $\hat{\...
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Convergence in probability does not imply convergence in $r^{th}$ mean

I am confused regarding convergence in probability and convergence in $r^{th}$ mean. I am able to prove that convergence in $r^{th}$ mean implies convergence in probability, which is not true. Let me ...
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rate of convergence acceptance-rejection vs inverse transform sampling

Does the acceptance-rejection method or inverse transform sampling converge to the mean quicker say for beta distribution, assuming acceptance-rejection has suitable envelope function? is there a way ...
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Show that $nX_{(1)}$ is not consistent

Consider a random sample from exponential distribution with mean $\frac{1}{\theta}$. I have to prove that $nX_{(1)}$ is not consistent for $\frac{1}{\theta}$ . A sufficient condition for consistency ...
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Can you use nAGQ = 0 with the Poisson distribution?

I am working with a GLM with lots of random variables and Poisson distribution. I get the error 'boundary (singular) fit: see ?isSingular' and so looked up ways around this. I found someone ...
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Limiting Distribution of $n\left[Y_n\right]$ where $Y_n$ is the minimum of a sample of size n from Uniform$\left(0,\theta\right)$ distribution

Suppose $X_1,X_2,\dots,X_n$ is a random sample from Uniform$(0,\theta)$ for some unknown $\theta > 0$. Let $Y_n$ be the minimum of $X_1,X_2,\dots,X_n$. (a) Suppose $F_n$ is the CDF of $nY_n$. Show ...
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Maximum likelihood convergence in mixture gaussian

Suppose there are two datasets $D_{1}$ and $D_{2}$ with same structure, which means the cluster and cluster proportion is the same. The only difference between them is that the size of $D_{1}$ is $n_{...
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Is it interesting to do several updates using the same batch in Stochastic Gradient Descent

I am working on a reinforcement learning problem. I was given a code where people used to train their neural-network as a Q-function estimator. During the training process, they sample $m$ (m small) ...
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Convergence rate of the inverse covariance matrix [closed]

I am trying to find results regarding the convergence rate of the inverse covariance matrix in the case where the number of observations $n$ is larger than the number of dimensions $p$. Assume that $...
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How to quantify rate of convergence in terms of number-of-observations instead of iterations?

I observe discrete points of data, and wish to compute an integral across those points. Since the data is quite sparse, I need to interpolate and extrapolate. There are various approaches in use (...
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Posterior convergence in expectation vs probability

Let's assume that we are doing approximate Bayesian inference and compute the convergence of our posterior estimate to the true value of the parameter using Wasserstein distance. Why posterior ...
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random walk on Z towards the origin

Consider a random walk on $\mathbb{Z}$ with rate $a>0$ (begin no origin). The r.w. jumps one step towards the origin with probability $p$ or one step away from the origin with probability $1 −p$. ...
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Showing $Y_n\stackrel{p}\to Z$ where $Y_n=B_nZ+(1-B_n)X$

I am reviewing some of my old class notes again, and I came across the following problem. I think I have solved the problem correctly, but I wanted to see what others here thought. Do you think I ...
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Regression coefficient convergence

Why does the term underlined in RED converge to the term underlined in GREEN? Can someone please provide a proof? Thanks in advance!
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Same Example for Two Counter Examples

I'm learning some probability theory and I've come across the following: For an example of a sequence of random vairables that converges in the mean square sense but not almost surely: We set $$P(X_n=...
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Linear Mixed Model Failing to Converge

I am attempting to run a Multilevel Mediation in R with overtime data (4 time points, 50 participants). I was hoping to create two new columns for each outcome and predictor variable, a baseline ...
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Convergence of covariance matrix

I was looking for a simple way to find the number of samples $n$ needed to get a decent approximation to the covariance matrix $\boldsymbol{\Sigma}$. Given a random sample $\{ \mathbf{X}_1,\mathbf{X}...
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When will positive sum of random variables converge to the sum of positive ones?

Suppose we have a sequence of identically distributed continuous random variables $x_1, \dots x_N$, with $x_i \in [0,C]$, and a constant $0\leq a\leq C$. We know by Jensen's inequality that $$\...
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Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$?

Suppose $(X,Y)$ has the pdf $$f_{\theta}(x,y)=e^{-(x/\theta+\theta y)}\mathbf1_{x>0,y>0}\quad,\,\theta>0$$ Density of the sample $(\mathbf X,\mathbf Y)=(X_i,Y_i)_{1\le i\le n}$ drawn from ...
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Momentum updates average of g, Adagrad also of g^2 - any other interesting updated averages for SGD convergence?

Updating exponential moving average is a basic tool of SGD methods, starting with of gradient $g$ in momentum method to extract local linear trend from the statistics. Then e.g. Adagrad, ADAM family ...
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Saddle-free Newton method for SGD - while Newton attracts saddles, is it worth to actively replel them?

While 2nd order methods have many advantages, e.g. natural gradient (e.g. in L-BFGS) attracts to close zero gradient point, which is usually saddle. Other try to pretend that our very non-convex ...
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1answer
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Understanding the infinite sum of random variables

I am doing a course on time series analysis, and am struggling with this definition: We call a weakly stationary process $\{X_t\}$ invertible with respect to a white noise $\{\epsilon_t\}$ if ...
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Does it make sense to scale categorical variables in glmer when they have three levels?

I am trying to fit a generalised mixed effects model, but I am having convergence problems. The model I want to fit is ...
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The IQs from 181 boys aged between 6-7 years old were measured. Calculate its mean's confidence interval for $\alpha = 5\%$

The IQ from 181 boys aged between 6-7 years old were measured. The mean IQ is 108.08, and the standard deviation is 14.38. (a) Determine the confidence interval with confidence coefficient $95\%$ ...
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Convergence in probability (asymptotic notation) result

Let $h=h_n$ be a sequence of numbers such that $h_n \rightarrow 0$ as $n \rightarrow \infty$, $\mu$ be a real constant and $f$ be some probability density function. I was wondering if the following ...
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Distribution of N objects into C bins that are then sorted?

Let's say we have $C$ bins and $N$ indistinguishable objects. For each object we choose one bin at random where each bin is equally likely (with probability $1/C$). Let $B_k$ be the number of objects ...
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Do fully connected layers in the middle of a network impede optimization?

I submitted a paper that uses an auto-encoder network with several convolutional layers in both the encoder and the decoder and a fully connected layer (FCL) in between. Besides the FCL being useful ...
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The limit distribution of Wilcoxon signed rank statistic?

An alternative representation of the Wilcoxon signed rank statistic $V$ is $V=\sum_{i\le j}\mathbb{I}_{\{X_i+X_j>0\}}=\sum_i\mathbb{I}_{\{X_i>0\}}+\sum_{i<j}\mathbb{I}_{\{X_i+X_j>0\}}$ ...
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1answer
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Convergence issues with lme4 1.1-20 for models that converged when using earlier version of lme4

I am encountering convergence problems with some models after updating to lme4 1.1-20 that I did not encounter with earlier versions of lme4 (in particular, lme4 1.1-15). I am encountering these new ...
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GLMM in R doesn't converge, nearly unidentifiable [duplicate]

I'm building my GLMM using r. ...
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Should one track the loss or accuracy of a neural network when training it?

Should one track a model's progress using its loss or its accuracy? I ask this because sometimes the loss at a epoch is higher than that at previous epochs (which is a bad thing) but so is the ...
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Are uniform LLNs preserved under monotone transformations

A simple question that I'm trying to answer is the following if I have a uniform LLN for a sequence of random vector; namely \begin{equation} \sup_\beta \left\| \frac{1}{n} \sum_n^NX_n(\beta)\right\| ...
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Is it the case/is there a proof that the convergence in distribution for the CLT is monotonic?

So for instance, if I compare $\bar{x}_n$ and the comparable normal distribution, and $\bar{x}_m$, $m > n$, and the comparable normal distribution, would I expect the difference in former (e.g. the ...