All Questions
34
questions
0
votes
1
answer
25
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
113
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 ...
- 2,612
0
votes
2
answers
65
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
$$...
- 13
0
votes
0
answers
55
views
Why is Convergence in Distribution defined in weak terms?
Why is convergence in distribution defined in terms of "weak" convergence in the law?
Intuitively, (at least at a literal level) convergence in distribution of $(X_n)_n$ sequence of Borel ...
- 593
0
votes
0
answers
76
views
Proving a (simple) step for the Skorokhod representation theorem [duplicate]
I am working on a slight variation of the proof of the Skorokhod representation theorem, as found in Bogachev, Vol. II. In particular, I want to use another definition of the functions $F_\mu,\xi_\mu$....
- 16k
0
votes
1
answer
28
views
Weak convergence of probability measures with fixed second moments
Consider Borel probability measures $\mu_n, \mu$ ($n \in \mathbb{N}$) on $\mathbb{R}$ such that
$\mu_n \to \mu$ weakly (test functions are continuous and bounded);
$E_{X \sim \mu_n}[X] = 0$ for each $...
- 1,692
3
votes
1
answer
625
views
Does weak convergence imply pointwise convergence? [duplicate]
Let $f_n,f:\mathbb R^d\to[0,\infty)$ with integral $1$ over $\mathbb R^d$.
Suppose that
$$ \int_{\mathbb R^d}\phi(x)\,f_n(x)\,d x\,\to\,\int_{\mathbb R^d}\phi(x)\,f(x)\,d x $$
as $n\to\infty$ for all $...
- 893
3
votes
1
answer
191
views
A tight collection of probability measures on space of probability measures
Let $E$ be a Polish space, and let $\mathcal M_1(E)$ denote the space of probability measures on $E$. I want to show the following:
A collection $\mathcal K \subset \mathcal M_1\left(\mathcal M_1(E)\...
- 4,075
7
votes
3
answers
952
views
Conditions for weak convergence and generalized distribution functions
I am having some trouble proving Corollary 6.3.2 in Borovkov's Probability Theory (for reference, this material is on pages 147 to 149 in the book). For convenience, I provide some definitions and ...
- 607
2
votes
0
answers
183
views
Weak convergence on a separable and locally compact metric space
Let $(E,d)$ be a separable and locally compact metric space. $(\mu_n)_n$ and $\mu$ are probability measures on $(E,\mathcal{B}(E))$, such that for all continuous function $f$ with compact support $$\...
- 155
3
votes
1
answer
202
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 ...
- 5,601
1
vote
1
answer
100
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
115
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
5
votes
1
answer
781
views
Necessary and sufficient condition for weak convergence and convergence of density
Let $(\mu_n)_n$ and $\mu$ be two probability measure, having respectively density $(f_n)_n$ and $f$ for the measure $\lambda$ on $(\mathbb{R},B(\mathbb{R})).$
Prove that the following statement are ...
- 1,070
2
votes
3
answers
130
views
Computing $\lim_{n\to\infty} \prod_{k=1}^n(1-\frac{x^2k^{2\alpha}}{n^{2 \alpha+1}})$
Let $\alpha>0,x \in \mathbb{R}$
I am having a problem in computing the following limit:
$$\lim_{n \to \infty} \prod_{k=1}^n\bigg(1-\frac{x^2k^{2a}}{n^{2a+1}}\bigg).$$
In fact: the problem was ...
user666433
1
vote
1
answer
194
views
Characteristic functions and metric spaces
Let $\mathcal{P}$ be the space of probability measure on $\mathbb{R}.$ Define $d(\varphi,\phi)=\sup_x|\varphi(x)-\phi(x)|/(1+|x|),$ where $\varphi$ and $\phi$ are the characteristic functions of two ...
- 616
6
votes
1
answer
1k
views
Lévy's metric on $\mathbb{R}^d$
I know that a sequence of measures on $\mathbb{R}$ converges in distribution if and only if the corresponding Lévy's metric converges (Relationship to weak toplogy (Lévy metric)).
According to ...
- 616
1
vote
2
answers
251
views
$P_n \Rightarrow P$ if and only if $P_n\{x\} \to P\{x\}$ for all $x$ if $S$ is a countable discrete space.
This is problem 2.3 from Billingsley's Convergence of Probability Measures.
If $S$ is countable and discrete, then $P_n \Rightarrow P$ if and only if $P_n\{x\} \to P\{x\}$ for each singleton. Show ...
- 5,601
2
votes
1
answer
74
views
How to show the following weak convergence using characteristic functions
Suppose $g:R\rightarrow R$ has at least three bounded continuous derivatives and let $X_i$ be $iid$ and in $L^2$. Prove that:
$\sqrt{n}[g(\overline{X_n}) - g(\mu)]\xrightarrow{w} N(0,g^{'}(\mu)^{2} ...
user621937
5
votes
1
answer
1k
views
How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric
The Lévy metric between distribution functions $F$ and $G$ is given by:
$$\rho(F,G) = \inf\left\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\right\}.$$
Another way to ...
user621937
3
votes
1
answer
1k
views
Convergence of Maximum of Cauchy Random Variables
Suppose $\{P_n\}$ and P are probability measures on the real line with corresponding distribution functions $\{F_n\}$ and $F$, respectively.
$P_n$ converges weakly to P if and only $$\lim_{n \...
- 401
2
votes
0
answers
394
views
Convergence of Probability Measures and Respective Distribution Functions
Suppose $\{P_n\}$ and P are probability measures on the real line with corresponding distribution functions $\{F_n\}$ and $F$, respectively.
Prove that $P_n$ converges weakly to P if and only $$\lim_{...
- 401
3
votes
1
answer
152
views
Trouble connecting pieces of proof in Kesten's seminal paper on Sinai's random walk
In Kesten's 1986 paper (Limit distribution of Sinai's Random Walk) we read:
The proof of this lemma uses the fact that the symmetric simple random walk when properly rescaled converges to the ...
- 6,121
1
vote
0
answers
1k
views
Tightness of normal distributions
Consider the $\mathcal{N}(\mu_n,\sigma_n^2)$ distributions, where the $\mu_n$ are real numbers and the $\sigma_n^2$ non-negatives.
A sequence of probability measures $(\xi_n)_{n \in \mathbb{N}}$ on $...
- 1,999
2
votes
1
answer
400
views
Convergence of measures of sets with measure zero boundary
Let $P_k$, $k\in\mathbb{N}$, and $P$ be probability measures on $\mathbb{R}^n$ equipped with the sigma-algebra of Borel sets and suppose that $P_k\longrightarrow P$ weakly.
Let $A\subseteq\mathbb{R}^...
- 8,803
2
votes
1
answer
360
views
Weak convergence of Cesaro sums
Suppose $\{X_n\}_{n \geq 1}$ is a sequence of random variables which converges weakly to some random variable:
$$ X_n \overset{w}{\longrightarrow} X $$
Question: what happens to the Cesaro sums of $...
- 2,244
17
votes
1
answer
4k
views
Confusion with the narrow and weak* convergence of measures
Think of a LCH space $X.$ Consider the spaces $C_{0}(X)$ of continuous functions "vanishing at infinity" and the space $BC(X)$ of bounded continuous functions. Consider as well the space of Radon (...
- 1,139
3
votes
0
answers
44
views
A basic question on spaces of probability measures
This problem is regarding the space of probability measures.
For $N \geq 1$, let $\{e_i^N(.), i\geq 1\}$ denote a complete orthonormal basis for $L_2[0,N]$. Let $\{f_j\}$ be countable dense in the ...
- 1,999