# Tagged Questions

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### How to prove that if $X_n \to X$ in distribution then $X_n = O_p(1)$

The title says it all: How to prove that if $X_n \to X$ in distribution then $X_n = O_p(1)$ My idea was to approach this with some measure theory: If not $X_n = O_p(1)$, then there exists an ...
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### Convergence almost everywhere?

Consider the following two statements about a random sequence $X_n$: (1) $X_n \stackrel{a.e.}{\rightarrow} X$. (2) $\mathrm{P}\{|X_n-X|>\epsilon, \ i.o.\} = 0, \ \forall \epsilon>0$. (a.e. ...
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### densities being absolutely continuous wrt Lebesgue measure

I'm reading an article with an assumption similar to: "The density $f(.)$ exists and is absolutely continuous with respect to Lebesgue measure". I don't understand this assumption because $f$ is not ...
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### How to understand $\frac{dP}{dQ}$, where $P, Q$ denote two distributions?

I am currently reading a paper named Estimating Individualized Treatment Rules Using Outcome Weighted Learning by Zhao et al., where they wrote an equation $$\frac{dP^D}{dP}=\frac{I(a=D(x))}{P(A=a)}$$ ...
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### One question regarding r.v independence

I just encounter independence in a Statistics course, I get stuck in this question for a long time..any help will be extremely appreciated. First one is, if $X_1, X_2, X_3...X_k$ (finitely many) are ...
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### Necessary condition for pairwise sufficient statistic [duplicate]

I'm struggling to prove the following. If $T:\left(X,\mathbf{A}\right)\rightarrow\left(Y,\mathbf{B}\right)$ is a pairwise sufficient statistic for a set $\left\{\mu_0,\mu_1,\mu_2\right\}$ of three ...
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### Is the limit of a sequence of B-measurable functions itself B-measurable?

Let $\left(\Omega,\mathcal{A}\right)$ be a measurable space and let $\mathcal{B}$ be a sub-$\sigma$-algebra of $\mathcal{A}$. Let $g,f_1,f_2, f_3,\dots$ be real-valued functions with domain $\Omega$ ...
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### Book on Measure Theoretic Statistics

I'm looking for a book, preferably a good one, on statistics from a rigorous, measure theoretic point of view. Ideally, this book should be introductory in nature and cover no more nor less than a ...
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### Time Series: existence of moments $\Rightarrow$ existence of distribution?

This might come to you as a bit silly, because normally we are used to the vice-versa question. But here is what I have: a nonlinear time-series model, for which I can derive by infinite backwards ...
Let $Q=\{q_1,\ldots,q_n\}$ ($n\in\mathbb N$, $n>1$) a collection of elements and $d:\,Q\times Q \longrightarrow \mathbb R^+$ a distance between a pair of elements (as a measure of similarity). ...