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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44 views

If $X_n\overset{p}{\rightarrow}0$, and $Y_n\overset{d}{\rightarrow}Z\sim Normal$, does $X_nY_n\overset{p}{\rightarrow}0$?

If $X_n\overset{p}{\rightarrow}0$, and $Y_n\overset{d}{\rightarrow}Z\sim Normal$, does $X_nY_n\overset{p}{\rightarrow}0$? According to Slutsky theorem, I can directly get $X_nY_n\overset{d}{\...
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Convergence of a stochastic process

Consider the discrete time random process $X_n,n\in \mathbb N$, with $$X_{n+1}=(1-K)\cdot X_n+K\cdot\frac{G_n}{c}\cdot X_n$$ where $G_n$ is a random variable with expectation $\mathbb E[G_n\mid X_n]=\...
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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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118 views

What is the limit of the random forest (or bagging) estimator?

I am looking for a proof or intuition as to why the absolute limit of a random forest estimator is the expectancy of a single tree (see citation below), i.e: $$ \hat{f}_{rf}(x) = \lim_{B \to \infty}[\...
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67 views

Markov Chain Limit Proof

First, sorry for my bad english. I am having trouble proving this exercise (it came from some notes I had back in university, I am studying for my masters next year). Let $X$ be an aperiodic ...
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41 views

Convergence in probability and variance

I have a statistic $\widehat{\sigma}$ that depends on data $(X_i,Y_i)_{i=1}^{n}$ with a known distribution, and I want to be able to say that $\widehat{\sigma} - \sigma \xrightarrow{p} 0$ (or $=o_p(1)$...
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Does $\hat \theta_n=\theta+O_p\bigg(\dfrac{1}{\sqrt{n}}\bigg)$ imply that $p_{\hat \theta_n X}(x)=p_{\theta X}(x)+O_p\bigg(\dfrac{1}{\sqrt{n}}\bigg)$?

Let $X$ be a random variable. Let $\theta$ be a constant, and let $\hat \theta_n$ be a set of normally distributed random variables that converge in probability to $\theta$, that is, $\hat \theta_n \...
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43 views

Weak Law of Large Number to Central Limit Theorem [closed]

Consider a distribution with unknown mean μ and population standard deviation σ=30. Using the Weak Law of Large Numbers, what is the minimum sample size in order to attain a probability of at least 99%...
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Convergence in distribution and Slutsky's theorem

It is known that from the CLT, if $X_i \stackrel{\text{iid}}{\sim} F$ for some distribution $F$ with finite variance, then $$\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - \text{E}[X]) \stackrel{d}{\to} N(0,\...
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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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48 views

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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168 views

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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Limiting Distribution Given Samples from Joint Distribution

Let $(X_i, Y_i)$, $1 \leq i \leq n$ be independent and identically distributed samples from a joint distribution $F (x, y)$. Suppose that $E[X^4], E[Y^4] < \infty$. Now define $\sigma_{XY} = E[(X - ...
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CLT with inconsistent estimator

So I have the OLS estimator that is inconsistent due to the mean independence assumption being violated. I'm asked whether $\sqrt{n}(\hat{\beta}-\beta)$ converges when the sample size $n$ goes to ...
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192 views

Prove that the MLE exists almost surely and is consistent

I need to show that given an i.i.d sample $X_1,\dots X_n$ arising from the model: $$\{f(x,\theta)=\theta x^{\theta-1}exp\{-x^{\theta}\},x>0,\theta\in (0,\infty)\}$$ that the MLE exists with ...
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Convergence in Distribution and Ordered Statistics [duplicate]

Let $X_1, X_2, \ldots$ be iid from Exp$(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$. Find the limiting distribution of $M_n = Y_1 - \theta\ln(n)$ and $T_n = nY_n$, ...
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If $X_n \sim \text{Beta}(n, n)$ Show that $[X_n - \text{E}(X_n)]/\sqrt{\text{Var}(X_n)} \stackrel{D}{\longrightarrow} N(0,1)$

Let $X_n \sim \mathbf{B}(n,n)$ (Beta distribution), with pdf $$ f_n(x) = \frac{1}{\text{B}(n,n)}x^{n-1}(1 - x)^{n-1},~~ x \in (0,1). $$ Knowing that $\text{E}(X_n) = 1/2$ and that $\text{Var}(X_n) = 1/...
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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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288 views

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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111 views

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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Consistent estimator of $p^2$

$(X_1, X_2,...,X_n)$ is a random sample of size $n$ from $Bernoulli(p)$ distribution. $S_n=\sum_{i=1}^nX_i$. I have to check whether $\frac{S_n(S_n-1)}{n(n-1)}$ is a consistent estimator for $p^2$. $...
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About the validity of two statements

Let $f(x)$ be some smooth univariate density, and let the leave-one-out Nadaraya-Watson estimator $\widehat{f}_{-i}(x)$ be defined as follows: $\widehat{f}_{-i}(x)=\frac{1}{(n-1)h}\sum_{j=1,j\neq i}^...
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111 views

Why taking an average makes convergence to zero faster?

Let $f(x,y)$ be some density, and let the leave-one-out Nadaraya-Watson estimator $\widehat{f}_{-i}(x,y)$ be defined as follows: $\widehat{f}_{-i}(x,y)=\frac{1}{(n-1)h^2}\sum_{j=1,j\neq i}^nK(\frac{(...
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How to mathematically describe convergence of sample density to population density?

Suppose we have a normally distributed random variable $X$ representing some population. Suppose we draw $n$ samples from the population: $$ (X_1,X_2,\dots,X_n). $$ This can be done using the ...
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1answer
266 views

Converge of Scaled Bernoulli Random Process

Suppose a random sequence is defined by $X_n := n B_n$, where $B_n$ is a Bernoulli sequence such that $\mathbb{P}(B_n = 1) = 1/n$. I am interested in the convergence properties of this random process ...
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What is the implication for convergence of MCMC if the Gelman-Rubin statistic increases again after initial convergence?

I am working on an MCMC sampler and in tests sometimes observe the following behavior of the Gelman-Rubin convergence diagnostic $R$. Across the first $n$ samples I estimate $R$ after every $i$ ...
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How is OLS estimator converging in quadratic mean equivalent to its variance matrix converging to $0$?

$\newcommand{\E}{\mathbb{E}}$ $\newcommand{\Var}{\text{Var}}$ $\newcommand{\b}{\beta}$ Sorry that my title is not clear (if there is any better suggestions, I will edit it as soon as I can) I want to ...
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Proving the almost sure convergence of the Kolmogorov-Smirnov test statistic

Context: I have spent the last few weeks thinking about how the central limit theorem is enunciated: if we have a set of i.i.d. random variables $X_1, X_2, \ldots X_m$ then $\frac{\sum_{i=1}^mX_i - E[\...
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1answer
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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: ...
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Does the Glivenko-Cantelli theorem work back and forth?

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(...
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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 ...
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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 ...
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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 $R^2$ ...
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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 ...
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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 ...
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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 ...
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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)|...
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39 views

sum of n Fisher information matrix divided by n

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 $$I_X = \sum_{i=1}^{n}I_{X_i},$$ ...

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