# 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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### multilevel modeling with lmer(): understanding failure to converge in a toy example

I am trying to get a deeper understanding of failures to converge in multilevel models that I estimate with lmer(). "Failure to converge" is vague; I want to be ...
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### Is this proof of convergence in probability to zero correct?

I want to show that $A=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(\widehat{B}_{i}-B_{i})X_i$ converges in probability to 0, where $B_i=E(C_i|Z_i)$ and $C_i$ is i.i.d. binary and $Z_i$ is a discrete random ...
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### Loss is stuck at 67% and wont converge even with large epoch and early stopping criterion [duplicate]

I am training a very simple 2D dataset with 2 features. Its tabular data and contains only numeric information. I tried using keras to train a neural network but the performance does not bulge. I ...
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### 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 ...
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### Asking for feedback on the application of a Central Limit Theorem

Let $\{X_{n,i}:1\leq i \leq d_n\}$ be a triangular array of mean zero random variables where $d_n$ is a positive increasing sequence ($d_n\leq n$). Under some conditions, a Central Limit Theorem ...
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### For arbitrary random variable $Z$, prove $P(\lvert Z1_{B^{c}}\lvert > \epsilon) \leq P(B^{c})$?

This question is asked to understand proof of Lemma 9.15 from Keener. For arbitrary random variable $Z$, show that $$P(\lvert Z1_{B^{c}} \lvert > \epsilon) \leq P(B^{c})$$ for event $B$ and ...
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### Deciding whether the autocorrelation plot shows a good sign of convergence?

I am wondering whether the autocorrelation plot (from MCMC sampling) shows a good sign of convergence when there is some autocorrelation until the 4~5th lags (at 1st lag 0.6, at 2nd lag 0.26, at 3rd ...
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### Convergence of Poisson Random Variable

For $n \in N$, if $X_n \sim Poisson(\frac{1}{n})$ then PT: 1. $X_n \xrightarrow[n\rightarrow \infty]{P} 0$ $nX_n \xrightarrow[n\rightarrow \infty]{P} 0$ It says $X_n$ converges to 0 in ...
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### Cross-correlations in digit distributions

Update on 2/29/2020. All the material below and much more has been incorporated into a comprehensive article on this topic. The question below is discussed in that article, entitled "State-of-the-Art ...
If I have $X_i$ being iid, and $E(X_i)=\infty$, how do I show that $\limsup \frac{X_n}{n}=\infty$ almost surely? I.e. how do I show $P(\limsup_{n\to\infty} (\frac{X_n}{n})= \infty)= 1$?
I set up a model, simulated some data and tried to infer the wanted parameter $\alpha$. However it seems that there may be no convergence to the true parameter (result is either $-\alpha$ or $+\alpha$)...