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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Slutsky's theorem and joint convergence

Consider $Z_i$ a random variable that converges in distribution to $Z$ where $Z$ is a standard exponential random variable and $R_i$ a continuous random variable that converges in probability to $0$. ...
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When the sample mean converges to the population mean, does the probability that the sample mean is equal to the population mean tend to 0?

Let $y_1, y_2, \ldots , y_N$ be arbitrary real numbers and suppose a process of simple random sampling without replacement that selects $n$ out of $N$ elements. Then suppose that these $N$ elements ...
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Simulated data for logistic regression

I used the code below to create the random variable x1 and binary variable y, and fit the regression with y and x1. My questions are: Why regression coefficient estimates are not close to 2 and 10 (...
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Does a misspecified model always have lower likelihood value than the correct model?

Suppose the true dgp is $$ x_i \sim d_1(\theta_1), \quad i=1,\ldots,N $$ where $d_1$ is some probability distribution with parameter(s) $\theta_1$, but I wrongly assume $$ x_i \sim d_2(\theta_2). $$ ...
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How to show that $X_n + Y_n \to X + Y$ holds in the $L^1$ norm?

Let $X_n$ and $Y_n$ be sequences of random variables. Show that $X_n + Y_n \to X + Y$ (1) $X_nY_n \to XY$ (2) If $\mathbb{P}(X=0) = 0, \; \frac{Y_n}{X_n} \to \frac{Y}{X}$ (3) are true for convergence ...
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Show the ergodicity of a random sum of ergodic processes

We say that a mean stationary stochastic process $(X_t)_{t \in \mathbb N}$ - i.e. $E[X_t]= \mu_X$ for all $t$ - is ergodic mean if \begin{equation}\tag{I} \frac 1 T \sum_{t=1}^T X_t \overset {pr} \...
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But what if the 2-th absolute moments converge in probability?

I'm trying to understand a kind of convergence. I had posted another question, but I think it got too polluted and I decided to delete it and simplify it a bit. We know that $X_n \to X$ in mean square ...
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The results from 2 programs are conflicting on convergence issues in my multivariate logistic regression, how do I deal with this?

Currently I am analyzing a dataset using logistic regression, I ran it in R using the glm function to run a multivariate logistic regression with 12 predictors. Some of these are quite collinear as ...
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Minimisation of KL divergence vs minimisation of empirical processes indexed by a metric space in MLE

I am trying to relate the interpretations of MLE as (1)minimisation of KL-divergence and (2)minimisation of empirical processes indexed by a metric space. Questions: Is it always true that maximum ...
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A question related to uniform asymptotic negligibility (UAN) assumption

The uniform asymptotic negligibility (UAN) assumption is well know in probability theory. In my case, I have a definition of (UAN) for MA processes. Let $(X_n)_{n\geq 1}$ a sequence of MA processes: $$...
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There is no topology on the space of random variables s.t. a.s. convergent seqs are the converging seqs?

The post When do we find convergence in distribution to independent variables? prompted me to review convergence in distribution. As I read on, I encountered a property about almost sure convergence ...
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The probability limit of the inverse of an infinite-dimensional matrix

I am considering a question regarding the calculation of the probability limit for a high-dimensional inverse matrix. Specifically, suppose that $A_n, B_n \in \mathbb{R}^{N_n \times N_n}$ where $N_n \...
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Reference request: Almost sure weak convergence

(Reposting from Math StackExchange) I've encountered the term "almost sure weak convergence" of empirical measures in several places but haven't been able to find a textbook reference. Could ...
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Is this Markov chain of a UAU-process (unaware - aware - unaware) convergent and why?

I am currently looking at a Markov chain of $UAU$-process on a uni-weighted undirected network. Where individuals are aware of certain arbitrary information or not. The individuals are the nodes of ...
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Stopping rule in the rlm() function for MASS package

When I looked at the source code for the rlm() function: ...
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Probability never stop tossing a coin [duplicate]

Assume you're tossing a fair coin; you win only if you get (i+1) consecutive heads right after (i) consecutive tails; what is the probability this game never stops? My attempt is as below: $P$(never ...
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Different convergence rate between two different data representations with same neural net

I am currently training a ResNet18 on audio data recordings using Mel Spectrograms which are of shape $(128,6025)$ and a different, less known audio data representation - the Multi-Resolution ...
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From univariate to joint convergence in distribution

Let $X_n \rightarrow_d X$ and $Y_n \rightarrow_d Y$ where $X$ and $Y$ are i.i.d standard exponential random variables. However, I do not have that for any $n$, $X_n$ and $Y_n$ are independent. Can I ...
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Central limit theorem for asymptotically i.i.d. random variables

I observe a sequence of r.v. $X_1, X_2, \dots$ where each $X_i$ is a function of the sample size $n$. When $n \rightarrow \infty$ I have the following result: $X_1 \rightarrow^d E_1, X_2 \rightarrow^d ...
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CycleGAN cycle loss

I was reading the paper of CycleGAN and I was trying to implement it. However, my models does not converge to any good solution whatsoever, and since I've checked the implementation many times, I ...
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Is true that the sampling distribution of $\ln \left(\chi^{2}\right)$ converges to normality much faster than the sampling distribution of $\chi^{2}$?

If true is the consequence true that $X \sim \chi^{2}(k)$ then $\sqrt{2 X}$ is approximately normally distributed with mean $\sqrt{2 k-1}$ and unit variance? Also true that If $X \sim \chi^{2}(k)$ ...
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Convergence of a function having a big summation at each sample

I have the following function. $$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$ Here, $U_m$ samples are random numbers coming from a Gaussian distribution $$U_m \sim \mathcal{N}(\mu_u, \sigma_u)$$ and ...
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Mixed model does not converge

I have a question regarding a mixed model I am using: In a study, participants have been presented with 40 different news article headlines and indicated for each headline whether they would share the ...
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Does log-rank statistic converge in distribution to a beta distribution or Pearson I distribution?

The logrank test statistic compares estimates of the hazard functions of the two groups at each observed event time. It is constructed by computing the observed and expected number of events in one of ...
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RNN on count time series [duplicate]

I am trying to predict the following count time series using RNN. X axis is in hours. Y axis is customer demand. I have already tried using other methods like stochastic models in tscount in R. I am ...
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Does convergence of $\sqrt{n}X_n$ to $N(0,1)$ in distribution implies $X_n \rightarrow 0$ in probability?

This question stems from the WLLN and the Central Limit Theorem. Suppose we have $n$ iid random samples $X_1,\ldots,X_n$ with common mean $\mu$ and finite variance $\sigma^2$. Then the sample mean $\...
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What does it mean for a sequence of random vectors to converge to a random vector?

I am reading about convergence of random variables from Wikipedia and I come across this. Note that the condition that $Y_n$ converges to a constant is important, if it were to converge to a random ...
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A question about the delta method in asymptotic distributions

I am reading up on the delta method from its Wikipedia page. Under the heading Univariate delta method the statement of the method is as follows: If $$\sqrt{n}[X_n - \theta]\xrightarrow{\text{D}} \...
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Limiting distribution of $G_n(X_n)$

Consider two sequences of random variables. At each point in the sequence $X_n \sim F_n$ and $Y_n \sim G_n$, and let $F_n(t)$ and $G_n(t)$ denote their respect CDFs. The distributions $(F_n, G_n)$ are ...
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Why does my mixed effects model fail to converge when fixed effects are added? How do I solve this problem?

I'm running a study in which participants rate the politeness of two different types of smiles (two levels: rewarding and affiliative) presented in three different situational contexts (three levels: ...
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Comparing forecast models for signs of conversion

I am trying to analyze two external forecast models for weather data that each generate hourly forecasts twice a day for one week ahead. Thereby I get a panel-like dataset, in which I am interested in ...
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The convergence of random variables to standard normal distribution

Let $V_s$ be $n\times s$ real matrix and consisting i.i.d $\mathcal{N}(0,1)$ random variables [*]. Suppose that $O_s^1$ is the orthogonal matrix, its first column being the normalization of the first ...
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Can you apply continuous mapping theorem for sequence of random variables that diverges to infinity?

Let there be a sequence of real-valued random variables, $\left\{X_n\right\}_{n \in \mathbb{N}}$, and suppose that either $X_n \overset{p}{\to} c < \infty$ or $X_n \overset{\textrm{a.s.}}{\to} c &...
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Rate of convergence of the eigenvalues of the samples covariance matrix

Assume $\{ x_i \}_{i=1}^n$ to be i.i.d. normally distributed with mean 0 and covariance matrix $\Sigma$. What can we say about the convergence of the eigenvalues of the samples covariance matrix $\...
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Normal density's rate of convergence to 0 as mean goes to infinity while x and standard deviation are fixed

Consider the density of the Normal distribution given by $$f(x; \mu, \sigma) = \dfrac{1}{\sigma\sqrt{2\pi}}\exp\left(-\dfrac{1}{2}\left(\dfrac{x - \mu}{\sigma}\right)^2\right)$$ It is obvious that, ...
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Procedures to show that a process is not ergodic

I'm trying to show that a certain process is not ergodic, but as I don't have much experience, I would first like to learn how to show simple cases. We know that if a discrete stochastic process is i....
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Why the series to the power of two differs from the expanded AR(2) equation?

I am trying to rewrite the series $\tilde{R}_{t}=\beta_{2}\left(\sum_{s=0}^{\infty}\frac{U_{t-s}}{\phi^{s+1}}\right)^{2}$ as an $AR(2)$ using lag operators $L$. I expand the series, define $\alpha=\...
3 votes
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Can a sequence of PMFs converge to a PDF?

Is there a meaningful sense in which a sequence of PMFs (of a corresponding sequence of real-valued random variables) can uniformly converge to a PDF? Intuitively, it seems like a strange question to ...
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Strong Law of Large Numbers related proof

I am trying to prove the following: So far I have used Kronecker's Lemma as such: \begin{equation} \tag{1} \text{Since } \sum_{i=1}^{\infty} \frac{\sigma_i ^2}{B_i ^2} < \infty, \text{ then, } \...
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Is sequence of probability mass functions always uniformly bounded

Say that we have a sequence of discrete random variables, $\left\{X_n\right\}_{n \in \mathbb{N}}$, which converges to a random variable, $X$, with a continuous distribution, e.g., the Normal (Gaussian)...
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Condition for the asymptotic non-zero point estimation of the variance

we know that a condition for a non-zero point estimate of the variance for a finite sample is that there exist at least two integers $i,j$ such that $X_i\neq X_j$. In other words $\frac{1}{n}\sum\...
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Convergence Rate of $t$ Test Statistic (Regression)

Consider a simple regression model, $y=\beta^Tx+\epsilon$, say using the cars dataset. We get the following summary: ...
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Are the number of failures in an almost surely converging sequence guaranteed to be finite?

If we have a sequence $X_n$ that converges almost surely, it makes sense to say that $\mathbb{P}(\lim_{n \to \infty} {X_n} = \mu) = 1$. However, does this imply that the number of failures, i.e., $|{...
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Approximate Bayesian Computation with some information about the posterior distribution

ABC method works as follows. At iteration $i \geq 1$ : Draw $\theta_i \sim p(\theta)$. Generate a sample $X^{(i)} \sim p(X|\theta = \theta_i)$. Accept $\theta_i$ if $\rho(S(X^{(i)}),S_{obs}) < \...
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Reporting summary statistics for stochastic optimization algorithms

In employing stochastic optimization for applied problems, one typically runs algorithms like simulated annealing and genetic algorithms multiple times to get a sense of overall variability. Based on ...
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Correctly specifying nested random effects and fixed effects with the same variable: how do I specify without running in convergence issues in lmer?

This question is a follow-up question on nested random effects and fixed effects in lmer from the following answer. https://stats.stackexchange.com/a/228814/257284 ...
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Gamma multilevel mixed-effects generalized linear model with random intercept and random slope does not converge

I would be super thankful your help with an issue I have with a multilevel mixed-effects generalized linear model that I'm trying to fit to my ecological momentary assessment data using Stata. I am ...
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Linear Mixed-effect Model Could Not Converge (an issue participants' coding?)

I ran a linear mixed-effect model with 'participants" and "PV" (phrasal words) as a random effect, and the context as the main effect. I found that the model could not converge after I ...
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Linear Mixed-effect Model Could Not Converge (after changing participants' coding)

I ran a linear mixed-effect model with 'participants" and "PV" (phrasal words) as random effect, and the context as the main effect. I found that the model could not converge after I ...
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Limit distribution of the joint distribution of maximum and minimum of a sequence of random variables

Assume we have a sequence $\mathsf{X}_1,\mathsf{X}_2,\mathsf{X}_3,...$ of iid random variables. Then the Fisher-Tippet-Gnedenko theorem shows that $$ \mathbb{P}\left(\frac{\max\{\mathsf{X}_1,\mathsf{X}...

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