All Questions
34
questions
0
votes
1
answer
24
views
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
105
views
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
0
votes
2
answers
64
views
$(X_n, Y_n) \to (X, Y)$ in distribution (Le Gall 10.6)
$$
\newcommand{\N}{\mathbb N}
$$
I am paraphrasing this textbook question slightly.
Question:
Let $(X_n)_{n \in \N}$ and $(Y_n)_{n \in \N}$ be two sequences of real
random variables, and let $X$ and $...
- 125
0
votes
1
answer
53
views
When does convergence in distribution imply convergence of integrals?
Suppose $f_n \rightarrow f$ in distribution, where $(f_n)_{n \geq 1}$ is a sequence of integrable functions. Also suppose $\sup_{n \geq 1} |f_n|$ is well-defined and measurable. Does this imply that
$$...
- 13
0
votes
0
answers
55
views
Why is Convergence in Distribution defined in weak terms?
Why is convergence in distribution defined in terms of "weak" convergence in the law?
Intuitively, (at least at a literal level) convergence in distribution of $(X_n)_n$ sequence of Borel ...
- 583
0
votes
0
answers
76
views
Proving a (simple) step for the Skorokhod representation theorem [duplicate]
I am working on a slight variation of the proof of the Skorokhod representation theorem, as found in Bogachev, Vol. II. In particular, I want to use another definition of the functions $F_\mu,\xi_\mu$....
- 15.9k
0
votes
1
answer
28
views
Weak convergence of probability measures with fixed second moments
Consider Borel probability measures $\mu_n, \mu$ ($n \in \mathbb{N}$) on $\mathbb{R}$ such that
$\mu_n \to \mu$ weakly (test functions are continuous and bounded);
$E_{X \sim \mu_n}[X] = 0$ for each $...
- 1,692
3
votes
1
answer
616
views
Does weak convergence imply pointwise convergence? [duplicate]
Let $f_n,f:\mathbb R^d\to[0,\infty)$ with integral $1$ over $\mathbb R^d$.
Suppose that
$$ \int_{\mathbb R^d}\phi(x)\,f_n(x)\,d x\,\to\,\int_{\mathbb R^d}\phi(x)\,f(x)\,d x $$
as $n\to\infty$ for all $...
- 893
3
votes
1
answer
190
views
A tight collection of probability measures on space of probability measures
Let $E$ be a Polish space, and let $\mathcal M_1(E)$ denote the space of probability measures on $E$. I want to show the following:
A collection $\mathcal K \subset \mathcal M_1\left(\mathcal M_1(E)\...
- 4,055
7
votes
3
answers
945
views
Conditions for weak convergence and generalized distribution functions
I am having some trouble proving Corollary 6.3.2 in Borovkov's Probability Theory (for reference, this material is on pages 147 to 149 in the book). For convenience, I provide some definitions and ...
- 607
2
votes
0
answers
181
views
Weak convergence on a separable and locally compact metric space
Let $(E,d)$ be a separable and locally compact metric space. $(\mu_n)_n$ and $\mu$ are probability measures on $(E,\mathcal{B}(E))$, such that for all continuous function $f$ with compact support $$\...
- 155
3
votes
1
answer
201
views
$P_{n}(A)\xrightarrow{n \to \infty} P(A)$ for any $P$-continuity set $A$ iff $F_{n}(x)\xrightarrow{ n \to \infty} F(x)$ for all continuity points
For a while, I have been strunggling to find why Levy's continuity Theorem would imply that:
$P_{n} \xrightarrow{\text{distribution}} P\iff \phi_{n}(t)\xrightarrow{n \to \infty}\phi(t)\; $
since Levy'...
- 998
2
votes
0
answers
59
views
Question on the weak convergence of measures implying convergence of integrals over some boundaryless set
This is a setting from Ken Iti Sato's Levy Processes.
Define $$g(z,x) = e^{i \langle z,x \rangle} -1 - i\langle z,x \rangle c(x)$$
where $c(x) = 1+o(|x|)$ as $|x|\to 0$ and $c(x)$ is some bounded ...
- 5,591
1
vote
1
answer
99
views
Limit of sum of Poisson distributed random variables
I will just recapitulate the complete problem first, then show my solution.
Problem
$\{X_{n}\}_{n\geq1}$ is a sequence of independent r.v.'s, $X_{n}\in\text{Po}(\mu)$ for each $n$. $N$ is independent ...
- 705
1
vote
0
answers
114
views
Interchange of weak limits without uniform convergence
I have a collection of real valued random variables on the same probability space indexed by $\mathbb{N}^2$, $\{X_{n,m}\}_{n,m\in\mathbb{N}}$. For each $n \in \mathbb{N}$, I know that $\lim_{m \to \...
- 2,693