# 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.

799 questions
Filter by
Sorted by
Tagged with
9 views

### Bootstrap: mixing independent and time-series data together

I have a very computationally heavy simulator (large-scale agent-based transport simulation), which usually takes up to 5 days of run time in a large computer. The results are probabilistic, so ...
1k views

### Why is the limit of a Chi squared distribution a normal distribution?

My professor claimed that $\lim_{p\to\infty}\chi^2_p$ has a normal distribution. The claim was made on the basis of the Central Limit Theorem: as $p\to\infty$, we have a Normal$(p\mu, p^2\sigma^2)$. I ...
136 views

### Show that the distribution of $\frac{1}{\sqrt{n}}\sum_{i=1}^n(X_i^2-3)$ is normal

Let $X_1,\ldots,X_n$ be i.i.d. variables with $\mathbb{E}[X_i]=0$ and $\mathbb{V}[X_i]=3$ and assume that $\mathbb{E}[X^4_i]<\infty$, show that $$\frac{1}{\sqrt{n}}\sum_{i=1}^n(X_i^2-3)$$ ...
28 views

174 views

### 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 ...
29 views

### 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 $\Sigma_{j=-\infty}^{\infty} |b_j| < \infty$ ...
181 views

### 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 ...
232 views

### 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 ...
117 views

392 views

### 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 ...
53 views

### 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 ...
250 views

### 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/...
14 views

85 views

### 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 $1 −p$. ...
36 views

### 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 ...
26 views

### 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=...
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}...