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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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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glmer model convergence question

We are working with a longitudinal dataset, with three variables: WAIP, BPSRRI and group. WAIP and BPSRRI are measured repeatedly for 10 times and group refers to the group assignment of our subjects ...
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23 views

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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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 ...
2 votes
1 answer
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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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1 answer
28 views

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 ...
1 vote
1 answer
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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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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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1 answer
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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 ...
2 votes
1 answer
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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 ...
2 votes
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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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34 views

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 ...
1 vote
1 answer
989 views

SPSS: GLMM and(adjusted) odds ratio

I am performing a retrospective study and the relative statistic analysis. I am studying the the risk factors for the occurrence of complications during medical procedures. I have 50 subjects ...
2 votes
1 answer
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Is there a statistic such that for large sample sizes $a_n (\hat{\theta} - \theta) \sim N(0, \Sigma)$ approximately but $a_n \neq n^{1/2}$?

Various central limit theorems are of the form $a_n(\hat{\theta}-\theta)\sim N(0, \Sigma)$ approximately as $n \to \infty$ and usually $a_n = n^{1/2}$. Are there central limit theorems for statistics ...
1 vote
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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 &...
3 votes
1 answer
323 views

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 ...
2 votes
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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 $\...
2 votes
1 answer
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Convergence in probability: Does squeeze theorem apply?

Does squeeze theorem apply in convergence in probability? My statistics reference (where it talks about convergence in probability and its condition) does not cite it (but does seem to apply it), but ...
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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=\...
4 votes
2 answers
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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....
2 votes
1 answer
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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 ...
4 votes
3 answers
5k views

Predictive Variance of a Gaussian Process

Suppose $f$ is a function of some variable say $x$ ($x$ could be multi-dim). Then the GP assumption is written as follows $$f∼GP(m,k)$$ where $m$ is the mean function and $k$ is the covariance ...
2 votes
1 answer
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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: ...
1 vote
1 answer
38 views

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)...
2 votes
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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, } \...
0 votes
1 answer
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Authors frequently mention the convergence of their reinforcement learning algorithms. Do they imply a local or a global convergence?

I frequently come across authors in reinforcement learning papers mentioning that some or the other algorithm converges. Do they mean a local convergence or a global convergence? What do they ...
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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\...
3 votes
1 answer
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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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28 views

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 ...
1 vote
0 answers
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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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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 ...
1 vote
0 answers
29 views

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}...
4 votes
3 answers
480 views

Central limit theorem seems counterintuitive given Law of large number

From what I understand, the Central limit theorem says the sample mean is distributed normally when sample number tends to infinity. However, the Law of large number says sample mean converges in ...
13 votes
3 answers
4k views

Understanding the concept of "Bounded in probability"

My statistics book defines the concept of "bounded in probability" in the following way: Definition 5.2.2 (Bounded in Probability). We say that the sequence of random variables $\{X_n\}$ is ...
2 votes
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Rate of convergence of Machine learning models

I am currently doing some work on the double debiased machine learning algorithm by Chernozhukov et al. 2016. They achieve $ \sqrt{N} $ rate of convergence for estimation of a treatment parameter. ...
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Multivariate analysis of sample mean and sample variance

$\{X_n\}$ Let be a sequence of iid probability vectors with mean vector$ \mu$ and variance-covariance matrix$ Σ$. In this case, sample variance and sample covariance are defined as follows $S_{n,j}^2=\...
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Establish convergence to normal area

I will try and make my question abstract, since I have two problems of the same overall type. I am given a time series $Y_t$, t=1,...,T. I know that this time series has a (slowly) changing ...
2 votes
2 answers
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What does $\hat{\theta}_n = \theta + O_p(n^{-1/2})$ mean in terms of rates?

For $\hat{\theta}_n = \theta + O_p(n^{-1/2})$ we have $$\hat{\theta}_n - \theta = O_p(n^{-1/2})$$ Therefore, we have for any $\epsilon > 0$, there exists a finite $M > 0$ and finite $N > 0$ ...
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Equivalent definitions of $ L_2$ convergence?

I have been reading up on the convergence of random variables, and I have come across two commonly given definitions of $ L_2 $ convergence: $ \|X_n-X\|_{L_2} \to 0:$ $(1):\left(E|X_n - X|^2 \right)^{...
19 votes
3 answers
1k 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 ...
3 votes
1 answer
695 views

Almost sure convergence

The problem (not homework) is Consider the probability space $([0,1], B_{[0,1]}, P)$ where $B_{[0,1]}$ is the Borel set and $P$ is Lebesgue measure on $[0,1]$. For any integer $n>0$, there exist $m$...
3 votes
1 answer
902 views

sample size Monte Carlo experiment

I'm studying Monte Carlo analysis but I find very counter-intituive the computation of the minimum sample size in order to reach a certain level of precision. As stated in Monte Carlo methods in ...

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