# Tagged Questions

Convergence generally means that a sequence of a certain sample quantity approaches a constant as the sample size tends to infinity.

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### What's the convergence rate in the context of convergence in probability?

A sequence $z_n$ with $\lim_n z_n = z$ is said to have $Q$-linear convergence if a constant $r\in (0,1)$ exists such that $\displaystyle |z_{n+1} - z| \leq r \, |z_n - z|$, where $r$ is called the ...
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### What do we know about the rate of convergence of the mean of RVs with infinite variance?

What, if anything, do we know about the rate of convergence of of the mean of identically distributed, independent or stationary random variables drawn from a distribution with a finite mean and ...
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### Problem on convergence of sequence of random variables

Given any sequence of random variables $\{ X_n \}$; how do I show that there exists a sequence of real numbers $\{ \alpha_n \}$, such that $\{ \alpha_n X_n \}$ converges in probability to 0 ?
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### Continuous uniform random variables convergence question

Let $X_1, X_2, \ldots$ be independent $U(0,2)$ random variables and let $$Y_n = \prod_{i=1}^n \, X_i \;.$$ How do I prove or disprove that that $Y_n$ converges to $0$ almost surely ?
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### HMM Training: Testing convergence

What is the best way to test for convergence while training an HMM? I understand that we need to iterate till the change in parameters ( transition matrix, emission matrix ) is less than the threshold....
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### When 2% of the Bayesian Model have not converged?

I have model with 20000 latent parameters, set up in a Gibb's sampler. 98% of the parameters and sometimes 99.5% of the parameters satisfy the Geweke convergence statistic, have low autocorrelation ...
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### Non-convergence in nonlinear regression

I've been studying a manuscript from the chemistry literature (not mine) that resorted to a trick to obtain convergence of a non-linear model fitted to experimental data. They wanted to estimate a ...
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### Is supremum continuous on D[0,1]? Needed for continuous mapping theorem

I am currently in the following situation: I have a sequence of processes which converges weakly to a limit process in D[0,1]. I want to show that the supremum of the processes converges as well to ...
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### Why exactly is it that $\sup E[M_n^2] < \infty \iff E[A_{\infty}] < \infty$

Probability with Martingales: About $(c)$ My understanding is that $(c)$ is equivalent to: $$\sup E[M_n^2] < \infty \iff E[A_{\infty}] < \infty$$ Since $E[M_n^2] = E[A_n]$, we have ...
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### Ideal Inertia for k-mean clustering convergence

Is there an ideal "inertia" for K-mean convergence. For example I'm trying to cluster to 64 clusters using sci-kit. the output is ...
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### Self study - Determining function codomain to study convergence

I'm having a serious problem in and old exam paper, specifically a question on convergence of random variables. Let $X$ and $Y$ be two i.i.d exponential random variables with parameter $\lambda$, so ...
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### Show estimate converges to percentile through order statistics

Let $X_1, X_2, \ldots, X_{3n}$ be a sequence of iid random variables sampled from an alpha stable distribution, with parameters $\alpha = 1.5, \; \beta = 0, \; c = 1.0, \; \mu = 1.0$. Now consider ...
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### What is a hard data set for support vector machines?

Suppose that: There are only numeric features in the data set. There is a binary target variable (since a general nominal case is resolved by 1vs1 or 1vsTheOthers approaches). I have noticed that ...
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### Problem of infinite or missing values in Hessian at convergence in the case of joint longitudinal and survival sub model

I'm estimating a Joint longitudinal and survival model using the JM package. The procedure works well for a subset of my data, but if I try to use the entire data set I get the following warning ...
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### convergence of geometric mean/harmonic mean

Does any one know papers regarding the convergence of geometric mean or harmonic mean in probability, parallel to central limit theorem?
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### Does the product of three Gaussian random matrices converge in distribution to a Gaussian?

Suppose we have vectors $u,v \in \mathbb{R}^r$ with and matrix $W \in \mathbb{R}^{r \times r}$ where all entries of $u,v,W$ are iid $N(0,1)$. Does the following hold? \frac{1}{r} u^...
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### ergodic theory for markov processes

For an ergodic Markov Chain $$\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]$$ where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
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I am trying to understand the definition of stable convergence in law. I have found the following definition. Definition. Let $Y_n$be a sequence of random variables defined on a probability space $(\... 0answers 57 views ### Markov model parameter concentration and Fisher Information Matrix For iid data, the posterior on the parameter $$p(\theta \mid x_{0:T}) = \prod_{t=0}^T p(x_t \mid \theta) p(\theta)$$ is known to become independent of the prior which is the Bernstein-von Mises ... 2answers 111 views ### Independence of reward and future state in stochastic process? Consider a Markov decision process in which we transition from state$s_t \rightarrow s_{t+1}$by taking action$a_t$, and then apply an update to a single entry from a table of$Q$-values based on a ... 0answers 20 views ### convergence of coordinate descent applied to lasso When using coordinate descent for solving a lasso regression, does normalizing the features impact the convergence rate? 2answers 181 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 ... 2answers 53 views ### What happens to integration over a term that converges to zero in probability? I have to do integration like this$\int h(x) [\hat{g}_n(x) - g(x)] dx$,where$\hat{g}_n(x)$is a non-parametric estimator of$g(x)$and$\hat{g}_n(x) - g(x) = o_p(1)$;$h(x)$is an arbitrary ... 1answer 56 views ### Convergence diagnostic of Markov chain that converge to uniform Let$\Omega$be a finite state space,$(X_t)_{t\in\mathbb{N}}$be a discrete-time Markov chain that converges to the uniform distribution, and$P$be its transition matrix. I'm looking for different ... 1answer 91 views ### Simple example wanted:$ X_n $converges to$X$in probability but not almost surely I'm looking for a simple example sequence$\{X_n\}$that converges in probability but not almost surely. The example I have right now is Exercise 47 (1.116) from Shao:$ X_n(w) = \begin{cases}1 &...
$X_n\sim Beta\left(\frac{\alpha}{n},\frac{\beta}{n}\right)$ with $\alpha>0$ and $\beta>0$. Does $X_n$ converges to a distribution? How do I approach to show that this converges to a distribution?...