All Questions
Tagged with probability-theory weak-convergence
837
questions
3
votes
2
answers
175
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Exercise 2.8 from Billingsley
In Billingsley's Convergence of Probability Measures, Exercise 2.8 asks:
Suppose $\delta_{x_n}\Rightarrow P$. Then $P=\delta_x$ for some $x$.
Here, $\delta_{x}(A)=\textbf{1}_A(x)$ is the unit mass. ...
- 45
0
votes
1
answer
23
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Convergence-determining class is a separating class
When I was reading Billingsley's book "Convergence of probability measures", it is claimed that "A convergence-determining class is obviously a separating class". But I don't ...
- 141
2
votes
1
answer
40
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Extracting a subsequence which converges in the $t$-Wasserstein distance?
Posted this to MathOverflow as well.
Assume that $\mu_n$ are probability measures on $\mathbb R ^d$ with finite moments of order $t$, and $\mu_n\to\mu$ weakly.
Clearly, $\int |x|^t d\mu_n(x)$ is a ...
- 481
4
votes
1
answer
38
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$X_n \rightarrow_d X$ and UI implies $\mathbb{E}X_n\rightarrow\mathbb{E} X$
Claim: Let $(X_i),X$ be real valued r.v.s. Then $X_n \rightarrow_d X$ and uniformly integrable implies $\mathbb{E}X_n\rightarrow\mathbb{E} X$
How can I prove this claim directly?
Here's how I proved ...
- 195
0
votes
0
answers
13
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Interchanging infinite sum and limit in distribution
I'm trying to do a proof for a project and I've run into the following problem.
For each $j$ consider a sequence $(Y_{j,n})_{n \in \mathbb{N}}$ of random variables such that the different sequences ...
- 376
4
votes
0
answers
54
views
Consistency of bootstrap estimator that is continuously $\rho_\infty$-Frechet differentiable
Theorem. Suppose $T$ is continuously $\rho_r$-Frechet differentiable at $F$ with the influence function satisfying $0<E[\phi_F(X_1)]^2<\infty$ and that $\int F(x)[1-F(x)]^{r/2}dx<\infty$. ...
- 2,224
0
votes
1
answer
65
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In this case does convergence of marginal distribution imply joint convergence in distribution
If $(S_n)_{n\geq 1}$, $A$ and $B$ (here we assume $A$ and $B$ are defined on the same probability space) are well-defined random variables, and $f$ and $g$ are two measurable functions. Then if we ...
0
votes
1
answer
57
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Show that $(X_n, Y_n) \stackrel d \to (X,Y).$
Let $\{X_n \}_{n \geq 1}$ and $\{Y_n \}_{n \geq 1}$ be independent random variables such that $X_n \stackrel d \to X$ and $Y_n \stackrel d \to Y.$ Suppose that $X$ and $Y$ are also independent. Then ...
- 535
1
vote
2
answers
54
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Does convergence in distribution implies imply this inequity?
Consider a sequence of random variables $\{X_n\}_n$. Given that
$$X_n - \sqrt{n} c \to N(0,1)$$ in distribution ($c>0$ is a constant number bounded away from zero).
Show that for any constant $t>...
- 453
2
votes
1
answer
105
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Show that there exists $x \in \mathbb R$ such that $\mathbb P (Y = x ) = 1.$
Let $\{x_n \}_{n \geq 1}$ be a sequence of real numbers and $\{X_n \}_{n \geq 1}$ be a sequence of random variables such that $\mathbb P (X_n = x_n) = 1,\ n \geq 1.$ Let $Y$ be a random variable such ...
- 2,522
2
votes
0
answers
27
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Convergence in Distribution of a Particular Sample Average
Suppose $g_{n}(\cdot)$ defined on $[0,1]$ converges in distribution to a continuous Gaussian process. Let $U_{1},...,U_{n}$ be i.i.d. random variables following $\text{Unif}[0,1]$. Allow $g_{n}$ to ...
- 135
1
vote
0
answers
22
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Total variation convergence in context of stochastic processes
Given a stochastic process $(X_t)_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$, with $\mu_t$ denoting the law of $X_t$ ($\mu_t=\mathbb{P}\circ X_t^{-1}$), the convergence in distribution
$$X_t~\...
- 169
0
votes
1
answer
48
views
Weak convergence of dependent variables
$X_n \xrightarrow[]{d} X$, $Y_n \xrightarrow[]{d} Y$ where $X \sim N(\mu_x, \sigma_x)$ and $Y \sim N(\mu_y, \sigma_y)$, but $X_n \not\!\perp\!\!\!\perp Y_n$. What do we need to analyze $(X_n, Y_n) \...
0
votes
0
answers
50
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Reference for a good multidimensional portmanteau theorem
I need references for a good n-dim portmanteau theorem that includes the following equivalent assertions:
The sequence $(X_n)_n$ converges in distribution to a random vector $X$ of $\mathbb R^\nu$;
$...
- 2,088
2
votes
1
answer
106
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Is it true that any sequence of random variables that converge in distribution is tight?
Let the sequence of real random variables $\{X_n\} \to X$ in distribution (but not necessarily in probability). Is it true that $\{X_n\}$ form a tight sequence, and if yes, how do we prove it?
So we ...
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