# 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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### Indicator function with almost sure convergence and BC lemma 2 exceptions

Would it be possible to define an indicator function $$Xn(A) \sim Bernoulli(\pi)$$ such that $\sum{\pi} = \infty$ and $$A\sim Uniform(0,1)$$ At the same time having $Xn(A)$ converging to $0$ almost ...
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### Proof of multivariate central limit theorem

$\newcommand{\phi}{\varphi}$ $\newcommand{\eps}{\epsilon}$ I'm using the book called 'A Course in Large Sample Theory' from Thomas S. Ferguson. During studying the proof of the central limit theory in ...
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### Interpretation of odd Central Limit Theorem (i.i.d) condition

My class was taught a third sufficient condition for the CLT to hold in the i.i.d. case that can replace the Lindeberg or Lyapunov conditions. I have never seen this condition before and am wondering ...
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### Mean square convergence of a series of stationary random variables

In Brockwell and Davis's book (Time Series Theory and Methods 2nd Edition), provide the following problem: Show that if $\{X_t, t=0, \pm1, \dots\}$ is weak stationary and $|\theta| < 1$ then for ...
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### Chains did not converge and mix well

I am trying to compute the parameter of copula model (Gumbel copula). I have used zero/one trick to write the copula likelihood model. The chains did not converge or mix at all. I have tried several ...
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### Convergence of policy iteration with discount $\gamma=1$

This question is related to my question (and my comment of RobPratt's answer) at https://math.stackexchange.com/questions/3860303/markov-decision-process-with-target-states-and-shortest-path-as-only-...
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### Lots of modes of convergence for random variables but why is there nothing on convergence of probability density functions?

There are numerous modes of convergence for random variables. But why do I never read anything about convergence of probability density functions? It seems like this would also be an important notion ...
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### Bounding the difference between the PDF of $\hat\beta\sim\mathcal{N}(\beta,\sigma^2)$ and a Dirac delta $\delta_\beta$ in terms of the variance?

Let $\beta \in \mathbb{R}$, let $X$ be a continuous random variable, and let $\hat \beta \sim \mathcal{N}(\beta,\sigma^2)$. Let the pdf of a random variable $A$ be denoted by $p_A$. Consider the ...
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### MCMC: long burn in vs re-initialization of the chain?

The developer of the well-known emcee package often gives this advice to help with chain convergence: Run a short (few hundred steps) chain Reinitialize all the ...
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### When do posteriors converge to a point mass?

What are the necessary conditions for a model's posterior to converge to a point mass in the limit of infinite observations? What is an example that breaks this convergence result? Off the top of my ...
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### Confidence Interval for Exponential Parameter Using Limiting Distribution

Suppose $X_1, X_2, \dots, X_n$ are iid samples from some $Unif(a, b)$ distribution, with $a < b$. Now let the random variable $Y_n = \min (X_1, X_2, \dots, X_n)$. Determine the limiting ...
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### Self-study. Does non-convergence in distribution implies non-convergence in probability?

This question arised after reading a solution to the problem. Given following problem: and it's solution: I may be wrong but it is not correct solution to a given problem. The fact that $X_n$ doesn'...
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### Convergence in probability and Chebyshev inequality

Given problem: The elegant solution is to use Markov inequality for $X^2_n$. But my solution was via Chebyshev inequality, smth like that: $P(|X_n - 1/n| \ge k) \le \frac{\sigma^2}{k^2}$, now lets ...
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### Suppose $\widehat{m}'(x)$ is the derivative of Nadaraya-Watson estimator, can I get its uniform rate from the rate for its numerator and denominator?

Suppose $E(Y|x)=m(x)$ is the regression function that is twice differentiable, $f(x)$ is the density of $X$ that is also twice differentiable. Suppose $Y_i=m(X_i)+e_i$. $m'(x)$ is the derivative of ...
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### Convergence in Distribution in Order Statistics

Let $X_1, X_2, \ldots$ be iid from Exp$(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$. (a) Find the limiting distribution of $M_n = Y_1 - \theta\ln(n)$ and $T_n = nY_n$,...
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### Convergence of function of central limit theorem

I have an estimator of the form: $$\frac{\hat{\mu}-c}{\sqrt{n\hat{\sigma}^2}}$$ Where $c$ is constant, $\hat{\mu}$, is an estimate, $n$ is number of observations and $\hat{\sigma}^2$ is a variance ...
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### Model not singular but doesn't converge what could be the reason (lme4 in R)

I'm following up on this great answer regarding running Principal Component Analysis (PCA) to uncover the reason behind lack of convergence and/or singularity for Mixed-Effects models. My model below ...
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### GLMM, repeated measures count data and Poisson regression: not positive definite, scaling, and convergence issues

Summary Using the lme4::glmer package I am trying to run a Poisson regression model with fixed effect, random intercept, and random slope. I have watched many tutorials and it seemed like this was ...
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### Weak convergence of moment generating function

I have the following sequence of rvs $$Z_1 = X_0*Y_0$$ $$Z_{n+1} = Z_n /2 + X_n*Y_n$$ Where $X_n$ and $Y_n$ are independent, with $X_n$ having Bernoulli distribution with p=1/2 and $Y_n$ having ...
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### Structural Equation Model no solution found

I'm trying to use the lavaan package in R for a SEM. I'm using ~30 variables. I scaled the variables with the scale() function. This is how my model looks like: ...
If we have a sample $X_1, X_2, \ldots, X_n \sim F$ then $\hspace{1mm}sup_x|F_n(x) -F(x)|\xrightarrow{a.s./p}0$. Now, if I can come up with a theoretical cdf $F$ such that $\hspace{1mm}sup_x|F_n(x) -F(... 1answer 35 views ### Notation on Resnick's for proving continuity of P and Fatou's lemma I'm using Resnick's "A probability path" and I'm bit confused with his notation (particularly regarding$\uparrow$and$\downarrow$)when proving the continuity of the measure P for monotone ... 1answer 58 views ### Not reaching convergence with mixed model I've got a study in which patients (record_id) can have from 1 to 5 aneurysms (concurrently) and each may be treated differently (each aneurysm). We are interested ... 1answer 88 views ### Convergence of random walk in$R^2$to the Brownian motion on circle We know that the random walk generated in$R^1$can converge weakly in distribution to the Brownian motion in$R^1$. Could anybody provide a mathematical proof, how a random walk generated in$R^2$... 1answer 126 views ### What does$P(|X_n - X| \geq \epsilon)$represent intuitively? I get that$P(|X_n - c| \geq \epsilon)$represents the probability that the random variable$X_n$is outside the interval of$(c - \epsilon, c + \epsilon)$but I am not sure how it works with a random ... 0answers 109 views ### Error in a zero-inflated negative binomial model? Following DHARMa diagnostic tests revealing zero-inflation (ratioObsSim = 32.663, p < 2.2e-16) and over-dispersion ... 1answer 93 views ### product of asymptotic standard normal distribution Suppose$Z_n\xrightarrow{d} Z \sim N(0,I_p)$, why$Z_n^TZ_n\xrightarrow{d}\chi^2_p$? I encounter this problem when we get the asymptotic distribution of the maximum likelihood estimator (MLE). Suppose ... 2answers 118 views ### asymptotic normality for MLE Suppose under suitable assumptions, $$[I(\theta_0)]^{1/2}(\hat{\theta} - \theta) \xrightarrow{d} N(0, I_p),$$ where$\hat{\theta}$is maximum likelihood estimator of$\theta$.$I(\theta_0) = I(\theta)|...
Suppose $X_i$'s are independent, where $i=1,2,...,n$. The density function of $X_i$ is $f_{\theta_i}(x)$. Let Fisher information of $X=(X_1, ...,X_n)$ be $I_X$. Then $$I_X = \sum_{i=1}^{n}I_{X_i},$$ ...