# Tagged Questions

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### Conditions for convergence of moments

Let ${X_n}$ be a sequence of r.v. such that $X_n\xrightarrow [d]{}X$, with $E(X)$ finite, and with $E(|X_n|^{1+\delta})\leq K<\infty$ for all $n$. We know that: a) For $\delta>0$, we have ...
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### convergence in probability of function of random variables

Suppose that $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d random variables. If we have $E(|X_1|^k) <\infty$ for some $k>0$ and $f(x)$ is a bounded continuous function on $\mathbb{R}$. Is the ...
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### Rate of convergence of mean in a central limit theorem setting

I recently asked a question here that was the following: If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that ...
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### What is the minimum standard deviation for a normal PDF such that one tail is always larger than that of a second normal PDF (different means)?

Say I have two weighted normal distributions, $$f_1(x) = \frac{a}{2 \sigma_1} e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}}$$ and $$f_2(x) = \frac{1-a}{2 \sigma_2} e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}}$$ ...
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### Convergence of sequences of random variables

Let $X_1, X_2, ...$ and $Y_1, Y_2, ...$ be two sequences of nonnegative random variables. Assume that each $n$ random variable $Y_n$ is uniform in the interval $[0, X_n]$. Show that if ...
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### Two different sequences of random variables each converge in distribution; does their sum?

My question is about basic probability. We have two sequences of random variables, $\{ X_n \}$ and $\{ Y_n \}$, such that each converge in distribution - i.e. there exist random variables $X$ and ...
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### Convergence in distribution and moments

Let us assume that we are given real random variables $X_n$ that converge in distribution to $X$. Moreover, it is known that $\sup_n \mathbb{E}[g(X_n)] < \infty$, where $g$ is a measurable function ...
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### Differentiability of random processes.

I know the appropriate criterions for mean-square differentiability of random processes. These criterions are connected with covariance function of a process. Are there any criterions for ...
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### Markov Chain Geometric Convergence

Perhaps due to my non-mathematician background (lack of Measure Theory knowledge), I have some difficulties about the Markov Chain Theory. Given an ergodic (irreducible and aperiodic) Markov Chain ...
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### Almost Sure convergence.

Given $(X_n, n\in\Bbb N)$ and $(Y_n, n\in\Bbb N)$ sequences of random Variables. For all $n\in\Bbb N$ it is : $X_n=Y_n$ almost sure. Now the question: Is then $P(X_n=Y_n \forall n\in\Bbb N)=1$? ...
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### Sufficient conditions for Uniform Law of Large Numbers

I would need a Uniform Law of Large numbers for $f_T(\theta)$ over $\Theta$ when $f$ is the indicator function and, thus, not continuous over $\Theta$. Do you know about any sufficient conditions?
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### Convergence of a matrix product

Let $A=o_{a.s.}(1)$; $A:k\times k$ matrix and $Vu=O_p(1)$; $V:k\times k$; $u: k\times 1$. Specifically, $Vu$ converges in distribution to $\mathcal N(0,I_k)$. Can we show that $VAu=o_p(1)$ or ...
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### Properties convergence of random variables

I want to understand the concepts of convergence of random variables better and therefore I wanted to find out how certain concepts that are straightforward for canonical convergence behave in this ...
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### Convergence rate of an estimator

Say we are interested in estimating some unknown real scalar parameter $\alpha$ using data. Suppose the estimator $\widehat \alpha_N$ of $\alpha$ using the data is consistent. I want to know what it ...
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### Convergence almost everywhere implies convergence in measure, the proof thereof

Let $(E, \mathcal{E}, \mu)$ be a measure space, and $(f_n)_{n\in\mathbb{N}}$ and $f$ be measurable functions $(E, \mathcal{E}, \mu)\longrightarrow (\mathbb{R}, \mathcal{B})$. The first part of Theorem ...
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### SLLN for nearly-identical observations

When observations $X_i\sim\mu$ are IID the usual SLLN along with separability of the underlying metric space yield that the empirical measure $\hat\mu_n=\frac1n\sum_{i=1}^n\delta_{X_i}$ converges ...
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### Almost sure convergence + convergence in distribution implies joint convergence in distribution?

I'm wondering, if I have two sequences of random variables $(X_n)$ and $(Y_n)$, defined on the same probability space, such that $X_n\stackrel{a.s.}{\rightarrow}X$ and $Y_n\stackrel{d}{\rightarrow}Y$, ...
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### Why is Expectation Maximization algorithm guaranteed to converge to minimum, even local?

I have read a couple of explanations of EM algorithm (e.g. from Bishop's Pattern Recognition and Machine Learning and from Roger and Gerolami First Course on Machine Learning). The derivation of EM is ...
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### Almost everywhere convergence of a function of rv

Let f(x) a continuous function on $\mathbb {\bar{R}^+}$(extended positive real line, $x\in(0,\infty]$). Take $y\in \mathbb R^+$. We can say that $\lim_{y\to 0}\frac{x}{y}=\infty$ almost everywhere ...
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### Uniform convergence in integrated survival function implies uniform convergence of distribution functions?

For a probability distribution function $F$ supported on a bounded interval $[a,b]$, the integrated survival function (ISF) is defined as $$\Psi_F(t)=\mathbb E_F\max\{X-t,0\}=\int_t^b(1-F(x))d x.$$ ...
Let $X_1, X_2, ...$ be a sequence of random variables such that: $X_i = \begin{cases} 1 &\mbox{with probability } p = 1-1/n^2 \\ 2 & \mbox{with probability } p = 1/n^2 \end{cases}$ and let ...