# 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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### 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 ...
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### If the weight and bias gradients are stuck at zero throughout training, is this an indication of dying ReLu?

A high learning rate when combined with a ReLu activation function is known to lead to the 'dying ReLu' problem. Is this a reasonable conclusion to arrive at if the gradient with respect to weights ...
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### Convergence in probability exercise

I have to solve the following exercise: Let $X_1,X_2,…,X_n,…$ be a sequence of random variables, with $X_n \sim \text{Uniform}(0,n)$. Also, consider $Y_n= e^{-X_n}$. Does the second sequence ...
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### Existence of Moments for Linear Regression With Pareto Error

Suppose I have the following model linear regression model: $y = \beta_0 + x_1i\beta_1 + x_2i\beta_2 + e_i$ with $e_i \sim Pareto(k,\alpha)$ Now if $1< \alpha < 2$, I would suppose that the ...
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### What is the big $O_p$ of the product between a $O_p(a_n)$ term and a uniformly bounded function?

Suppose $\frac{1}{n}\sum_{i=1}^n \hat{\theta}_i^2 = O_p(a_n)$ and $||f(X)||_{\infty}$ is bounded. What is the big $O_p$ of $\frac{1}{n}\sum_{i=1}^n (\hat{\theta}_i f(X_i))^2$? The way I understand ...
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### When investigating Monte Carlo convergence, should I reuse previous data?

I am doing Monte Carlo simulations and I want to investigate the convergence. Two versions come to my mind: 1) Doing every trial independently: For each trial, I generate new data independent from any ...
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### Multilevel (lmer) problems when using weights

I'm having some troubles when running my multilevel model. Specifically, I'm just fitting a starting simple multilevel model with time (2 years, level 1) nested in individuals (level 2). Here a sample ...
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### Product of a sequence of RVs (convergence in distribution) and another sequence (convergence in probability)

Suppose that $\{X_n\}\xrightarrow{d} X$ and $\{Y_n\}\xrightarrow{p} 0$. Now assume that $X\sim N(\mu,\sigma^2)$. I think that the following holds $$\{X_nY_n\}\xrightarrow{p} 0$$ My intuitive proof: ...
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### Bootstrap confidence interval for mean

Take an iid sample $X_1,\dots,X_n$ from a distribution with (unknown) mean $\mu$ and variance $\sigma^2$. Let $\bar{X}_n^b$ be the bootstrap sample mean, and choose $R_n=R_n(X_1,\dots,X_n)$ s.t. C_n=...
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### Accepting Parameter Estimates from Optimization Algorithms that have not Converged

Is there a popular consensus on accepting the solution from an optimization algorithm that has not converged? Suppose you let an optimization algorithm (e.g. gradient descent, Bayesian optimization) ...
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### How should a good cross-validated model look like? [closed]

I am baffled if i should make my model converge at every fold or let it converge at the last fold. Can someone explain me what path should i choose (for a regressional model)? My current "best&...
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### Convergence in Quadratic Mean Implies Convergence in Probability Proof Clarification

Wasserman (2004) All of Statistics A Concise Course in Statistical Inference gives the following proof that convergence in quadratic mean ($L_2$ convergence) implies convergence in probability: By ...
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### Probability of correctly identifying the Bayes class

Consider $X$ being a random variable taking value $\{1, \ldots, K\}$, with probability $p_1 = \frac{1}{K} + \varepsilon$ and $p_k = \frac{1}{K} - \frac{\varepsilon}{K-1}$ for all $k \neq1$. Taking i.i....
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