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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Convergence of estimated Survival Functions
Q1 part A&B I have so far
$$\underset{n\rightarrow\infty} {\lim} \frac{1}{n}\sum_{i=1}^nI(T_i>x)$$
since we are summing an indicator variable we can say it has a Bernoulli distribution with ...
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Is convergence in probability implied by consistency of an estimator?
Every definition of consistency I see mentions something convergence in probability-like in its explanation.
From Wikipedia's definition of consistent estimators:
having the property that as the ...
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How can we maintain asymptotic normality with slight change?
If $(X_n-\mu_n)/\sigma_n\rightarrow_{d} N(0,1)$ (i.e., $X_n$ is $AN(\mu_n,\sigma_n^2)$), I want to show the following two statements:
(1) $X_n$ is $AN(\bar{\mu}_n, \bar{\sigma}_n^2)$ if and only if $\...
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Strong consistency of kernel density estimator
I am studying the book Nonparametric and Semiparametric Models written by Wolfgang Hardle and have difficulty with the following exercise:
$\textbf{Exercise 3.13}$ Show that $\hat{f_h}^{(n)}(x) \...
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Convergence in Logistic Regression
Hey I'm taking a deeper dive into logistic regression. Specifically the following loss function with L2 regularization,
$$l(w)=\frac{1}{n}\sum_n \log(1+\exp(-y_i \cdot x_i^Tw))+\frac{\lambda}{2}||w||^...
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Seeking Feedback on Mixed-Effects Model with Weighted Observations
I've been working on a project where I'm comparing different methods using a mixed-effects model, and I'd appreciate some feedback on my approach.
Background:
I have a dataset with several variables:
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In case of no correlation, can a model make predictions above the expected values?
For simplicity's sake, let's suppose a binary classification problem, with a perfect 50% of probability for each of the classes, and a SkLearn's SVC model.
Let's ...
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Will this converge to origin?
Suppose you have a diffusion of 100 points with the following iteration:
$$(x_{n+1},y_{n+1}) \sim \mathcal{N}\left((x_n,y_n), \frac{x_n^2 + y_n^2}{2} I_{2 \times2}\right)$$
This will make a high ...
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Convergence of the SARSA algorithm
I'm trying to figure out the convergence of the SARSA algorithm, but I need help. In the article "On the Convergence of Stochastic Iterative Dynamic Programming" by Jakkola, Jordana and ...
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Convergence of $E(|X|^r)^{\frac{1}{r}}$ [closed]
For a random variable $X$ on $[0, 1]$ with $F(1) = 1$ and $F(x) < 1$ for all $x < 1$, show that $E(|X|^r)^{\frac{1}{r}} \to 1$ as $r → ∞$. If $F$ is such that $F(x) < 1$ for all $x ∈ \mathbb{...
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Is there a law or theorem related to occurrence of an event with highest probability in a population with infinite size?
Assume, we have a key that appears in either of the three rooms randomly (red room, blue room, and green room). We have the following probability distribution:
...
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Can't fit glmer in R with 18,000 observations when two proportions are very close
I have a dataset that has 3 levels (institution/provider/1:1 matched pairs) with 18,000 observations.
The matched pairs (subclass) were exactly matched within their institution and might receive ...
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Does this random sequence converge almost surely or in distribution?
In professor Leon Garcia's textbook "Probability, Statistics and Random Processes for Electrical Engineering"
there is an example (example 7.20 on page 383) of almost sure convergence of a ...
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What is the asymptotic bound for the ratio of sample mean and expectation？
For an i.i.d. observations $X_1,\cdots,X_n$ (bounded), we have the Hoeffding's inequality that establishes the upper bound for the tail probability of $|\bar{X_n}-\mathbb{E}[X_1]|$. I would like to ...
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Proof for equivalence definition of convergence in the Wasserstein space
From "An Invitation to Statistics in Wasserstein Space" (Victor M. Panaretos and
Yoav Zemel)
Theorem 2.2.1 (Convergence in Wasserstein Space) Let $\mu, \mu_n \in \mathscr{W}_p(\mathscr{X})$. ...
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$F_n(x) \to G(x)$ weakly if and only if $F_n(x) \to G(x)$ for all $x$ that is continuity point of $G$ and $0<G(x)<1$?
If $F_n$, $G$($G$ is non-degenerate) are distribution functions in $\mathbb{R}$, is the fact that $F_n \to G$ weakly($F_n(x) \to G(x)$ for all continuity point of $G$) equivalent to the condition that ...
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Understanding Why (or Why Not) a T-Test Require Normally Distributed Data? [duplicate]
This is a concept that I have always struggled to understand: We can write the formula for a Two Sampled T- Test (https://en.wikipedia.org/wiki/Student%27s_t-test) to compare the sample averages from ...
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How to visualize the proximity between two stochastic processes?
I am studying a certain convergence of time series. And I would like to "make sure" that the convergence to a certain process actually happens. So, consider the following simplification.
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A problem of weak convergence of stochastic processes
Let $(X_1,X_2, ....)$ be a infinite sequence of random variables. Supoose that te sequence is strictly stationary. For all $n$, define the following infinite array:
$$
\begin{bmatrix}
Y_1 & Y_2 &...
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Finite bracketing integral implies convergence in probability
In a paper I am reading, the following result is used. If $(\mathcal{X},\mathcal{F}) $ is a measurable space, $\mathcal{G}$ is a class of functions with elements $f : \mathcal{X} \to \mathbb{R}$. We ...
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Conditions needed for the convergence of Bayesian posterior distribution to point mass (posterior consistency)?
The following 2 theorems (from Bayesian Data Analytics 3rd edition by Gellman, appendix B) show proofs for why Bayesian posteriors converge to a point mass around θ0. Where θ0 is the true parameter ...
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Convergence of Sample Distributions for Periodic vs Chaotic Systems
Say we have a stochastic process from which we take finite samples to build up some sample probability distribution. As we include more time steps, the sample distribution converges to that of the &...
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A question related to the convergence of series in probability
Let $(X_j)_{j\in \mathbb Z}$ be an strictly stationary sequence of random variables with:
$$E[X_j]=0, \quad E[X_j^2] < \infty$$
I want to show that for each positive $\varepsilon$:
$$
\sum_{n=1}^\...
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Does convergence of moment implies existence of finite moment? [closed]
Let $X_n$ by a random variable that converges in distribution to $X$. On the top of that,
$$\lim _{n \rightarrow \infty} E[X_n] = E[X] = O(1).$$
Does it implies that for every $n$
$$ E[X_n] < \...
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Asymptotic distribution of $n^r \frac{U_{(1)}}{U_{(n)}}$: figuring possible $r>0$
Consider the i.i.d. sample $U_1, U_2 \cdots, U_n$ from the uniform distribution $U(0, 1)$. I should find a possible values of $r>0$ to have an asymptotic distribution of
$$ n^r \frac{U_{(1)}}{U_{(n)...
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Lower bounds for non-parametric regression in sup-norm with bounded noise
It is well-known that lower bounds on the nonparametric regression convergence rates under Gaussian (and more general unbounded) noise assumptions are given by $n^{-\frac{\beta}{2\beta + d}}$ in the $...
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Implementing Lebesgue's Dominated Convergence Theorem [closed]
I have to solve this limit using the Dominated Convergence theorem ( https://www.math3ma.com/blog/dominated-convergence-theorem ).
I'm not sure what to do with that "n" before the integral.
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Rewriting the expectation of f(x) by means of its derivative
I have a question regarding this proposition.
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an a.e. differentiable function so that $\int \frac{\left|f^{\prime}(x)\right|}{(1+|x|)^s} d x<\infty$ ($...
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non-reversible (irreversible) MCMC Langevin sampling methods. Intuition
Mathematically it is well understood that irreversible samplers are superior to reversible samplers in a number of ways. For example, if we want to sample from a distribution $\pi \propto e^{-U}$ by ...
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Rate of convergence of covariance of functional
Consider $X_n$ and $Y_n$ to random variable that are bounded in probability.
I know that
$$cov(X_n, Y_n) = O(n^{-1})$$
and that
$$(X_n, Y_n) \rightarrow_d (E_1, E_2)$$
where $E_1$ and $E_2$ are two ...
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Convergence of a subsequence of arrays of random variables
Consider $X(ij)$ for $i = 1, ..., n$ and $j = 1, ...,n$ be random variables.
I proved that for each i, the sequences $X($i$j)$ converge in distribution to random variable Y as $n$ tends to $\infty$. ...
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If $F^n(b_n x) \to e^{-x^{-\alpha}}$, $b_n x \to x_0$ where $x_0 = \sup \{x \colon F(x) < 1 \}$
Let $X_n$ be i.i.d with common df $F$. Let $M_n = \max (X_1, \ldots, X_n)$. Suppose $P(b_n^{-1} M_n \leq x) = F^n(b_n x) \to e^{-x^{-\alpha}}$ weakly, where $x > 0$ and $\alpha > 0$.
Let $x_0 = \...
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Does the following distribution converge to anything?
Consider the following process for generating a random sample:
Sample $X_1, X_2, \dots, X_n \sim \mathcal{N}(0,1)$
Compute $M = \max\limits_i |X_i|$
Scale the values to get $Z_i = X_i / M$
Can we ...
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Example of product of sequence of random variables that does not converge in distribution to the product of the limits
One result of Slutsky's theorem is that when $X_{n}$ is a sequence of random variables converging to a random variable $X$ and $Y_{n}$ a sequence of random variables converging to a constant $c$ then ...
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What does the likelihood function converge to when sample size is infinite?
Let $\mathcal{L}(\theta\mid x_1,\ldots,x_n)$ be the likelihood function of parameters $\theta$ given i.i.d. samples $x_i$ with $i=1,\ldots,n$.
I know that under some regularity conditions the $\theta$ ...
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Generalised speed measure balances between entropy of proposal density and average acceptance
This is in reference to page 3 of https://arxiv.org/abs/1911.01373
In the following line after equation (3): the author mentioned that the generalised speed measure (a measure of speed for which a ...
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$\mathbb E[|X_n|^r]<\infty$ and $\mathbb E[|X_n|^r]\to \mathbb E[|X|^r]$ as $n\to \infty$
Let $\{X_n\}\xrightarrow{d}X$ and for some $p>0$, we have
$$\sup_{n\ge 1} \mathbb E[|X_n|^p]<\infty$$
Show that for any $r\in (0,p)$, we have
a. $\mathbb E[|X|^r]<\infty$
b. $\mathbb E[|X_n|^...
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Checking the Lindeberg Condition
Let $\{X_n\}$ be a sequence of independent random variables such that $\mathbb P(X_n=\pm 1)=\frac 14$, $\mathbb P(X_n=\pm n)=\frac 1{4n^2}$ and $\mathbb P(X_n=0)=\frac 12 - \frac 1{2n^2}$ for all $n\...
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General Non-Uniform Berry-Essen
Let $f_n(x)$ by the probability distribution function of a continuous r.v. $X_n$. $X_n$ converges in distribution to $X$, i.e. $|P(X_n < x) - P(X < x)| \rightarrow 0$. On the top of that, $E[|...
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What is the definition of convergence in the context of deep neural networks?
Suppose I have a feed forward neural network which approximates a value, say $Y_0$. The analytical value of $Y_0$ is given. The plot of the network approximation of $Y_0$ each step is given as follows....
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Convergence in parameter implies convergence in distribution
I'm interested in the following question. It seems pretty elementary but I don't know where to actually find reference on it.
Suppose we have a scale (one parameter) distribution family $\{\mathcal{F}...
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$\limsup_n \dfrac{X_n}{n} = 0$ if $\mathbb{E}(X_1) < \infty$?
Here is an exercise in the book of author Achim Klenke. Let $(X_n)$ be iid non-negative random variables. By using Borel-Cantelli lemma, show that:
$$
\limsup_n \dfrac{X_n}{n} = 0 \text{ a.s}
$...
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Adding interaction effect in glmer model leads to convergence issues and unreasonable results
I am attempting to run a Mixed-Effects Logistic Regression in R, using the glmer function to predict the choice (0 = reject vs. 1 = accept) of a certain offer in an experimental task on a trial-by-...
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Compare an $O_p(1)$ with some growing numbers
There is a known positive sequence $M_n>0$ (e.g., $M_n = \log(n)$), where $M_n\to\infty$ as $n \to \infty$. If a sequence $X_n$ is $O_p(1)$, then can I claim $\Pr(|X_n| > M_n ) \to 0$ as $n \to \...
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Convergence almost surely of Bernoulli distributed random varibles
I proved that given a sequence of Bernoulli distributed random variables $ X_n$ with parameter $1/n$ they do NOT converge to $X=0$ a.s. using the Borel-Cantelli lemma.
My doubt is: If I have the space ...
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Nonconvergence of some parameters in MCMC of Hierarchical Bayesian Model
In short:
MCMC is used to construct posterior distributions for parameters of central tendency and all parameters used in the formula for this central tendency. I only care about the parameters of ...
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Convergence criteria for random field
I am iteratively solving a stochastic equation by generating a random field and using the resulting generation to move toward an equilibrium. I know that the system converges but I want to use an ...
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Proving uniform convergence of moment restriction score function in GMM asymptotic normality proof
I am asked in a homework question to prove asymptotic normality for the generalized method of moments estimator. The assumptions (which i think are necessary to solve this particular subproblem) given ...
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Why don't we typically drop a category as a baseline in Bayesian hierarchical linear regression?
Let's say we have two categorical variables the first with categories $j = 1,..., J$ and the other with categories $k = 1,...,K$. Often in Bayesian hierarchical linear regression, we might have a ...
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If an estimator for a parameter is unbiased, is it necessary for its variance to tend towards 0 for it to be consistent?
I have been taught that if an estimator is unbiased, then its convergence in probability can be proven by taking the limit of its variance as the sample size grows to infinity and showing it is equal ...
