# 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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### Expectation of 500 coin flips after 500 realizations

I was hoping someone could provide clarity surrounding the following scenario. You are asked "What is the expected number of observed heads and tails if you flip a fair coin 1000 times". Knowing that ...
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### Convergence in distribution of parameters of exponential family

I am taking a course in inference where we have to find an approximate confidence interval for a Rayleigh distributed variable. The correct answer to this question states: Since we have an ...
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### Asymptotic normality: proof strategy

Given a estimator $\hat \theta$ of $\theta$, I want to show that $\sqrt{n}(\hat\theta -\theta-B)\to N(0,V_\theta)$ as $n\to\infty$, given that the limit $V_\theta$ exists and $B>0$ possibly ...
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### Find the value of $\nu$ so that $n^\nu (1-X_{(n)})$ converges in distribution

Let $X_1, X_2, \cdots$ be iid. If $X_i \sim Beta(1,\beta)$, find the value of $\nu$ so that $n^\nu (1-X_{(n)})$ converges in distribution. My thoughts: Since $X_{(n)} \to 1$ in probability, I was ...
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### Convergence of series of dependent random variable, central limit theorem

My friend and I have a problem on central limit theorem. Given $X_1,X_2......$ are i.i.d random variables with mean $\mu$=0, variance $\sigma^2=1$(may or may not be normally distributed). If we ...
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### What is the relationship between Validation loss and Training loss when considering Overfitting? [duplicate]

Here I have results from my training stage I have been told that this would not be considered as overfitting, however, it seems the line follows the dots well and the validation loss is higher than ...
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### Clarification about the limiting distribution and approximate distribution of $\bar{X}^3$ using the delta method

The question states: Let $X_1,....X_n$ be a random sample from $f(x$,$\theta$) with $E(X$) = $\mu$ and $V(X) = \sigma^2$. Find the limiting distribution of $\bar{X}^3$, and the approximate ...
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### 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 ...
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### 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 ...
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### 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 $$\frac{1}{\sqrt{n}}\sum_{i=1}^n(X_i^2-3)$$ ...
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### Which is the better estimator for standard deviation?

Let $X_i \sim^{\textrm{iid}} N(\mu, \sigma^2)$. If I have measured $n$ values of $\textrm{std}(X_i)$ as $\sigma_1,\cdots,\sigma_n$, then what is the better estimator for $\sigma$: \hat{\sigma}_1 = ...
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### Probability of an event with probability 0 happening at least once in infinite trials

This question here is confusing me a lot. To summarize, let's say you have $\text{i.i.d. }X_i \sim U(0, 1), i = 1,2,\ldots, n.$ The question shows that $Y_i = \max(X_1, \ldots, X_i) \rightarrow 1$ ...
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### How can we conclude that an optimization algorithm is better than another one for a problem at hand

When we test a new optimization algorithm for a particular problem at hand, what the process that we need to do?For example, do we need to run the algorithm several times, and pick a best performance,...
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### Proving Linear Regression with Gradient Descent Converge to OLS estimates

Problem I am having trouble showing that the parameters $\theta\in \mathbf{R}^{m}$ for Linear Regression converge to the classic OLS estimates using gradient descent. Please find below my attempt: ...
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### convergence in distribution when parameters converges almost surely

let $X_n$ be sequence of random variables with the associated distribution $N(\mu_n,\beta_n+E)$. That is a sequence of normally distributed random variables with changing mean and variance. $\beta_n$ ...
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### How many iterations are too many?

I have the following model: ...
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### Issue with proof in Statistical Theory related to Beta and Binomial distributions [duplicate]

Assume $X_n$ is distributed $\text{Beta}(1/n, 1/n)$ and $X$ is distributed as $\text{Binom}(1,1/2)$. Show that $X_n$ converges to $X$ in distribution. I'm having some issue with this question. I ...
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### convergence and efficiency of mcmc chains and estimation of covariance matrix

I am doing some bayesian analysis and exploring posterior distribution with mcmc method. I would like some clarification with estimating the covariance matrix. I have a model with 6 parameters. ...