All Questions
34
questions
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
15
votes
1
answer
5k
views
Uniform convergence and weak convergence
Assume $F_{n},F$ are distribution functions of r.v.$X_{n}$ and $X$, $F_{n}$ weakly converge to $F$. If $F$ is pointwise continuous in the interval $[a,b]\subset\mathbb{R}$, show that
$$\sup_{x\in[a,b]}...
- 279
7
votes
3
answers
952
views
Conditions for weak convergence and generalized distribution functions
I am having some trouble proving Corollary 6.3.2 in Borovkov's Probability Theory (for reference, this material is on pages 147 to 149 in the book). For convenience, I provide some definitions and ...
- 607
6
votes
1
answer
1k
views
Lévy's metric on $\mathbb{R}^d$
I know that a sequence of measures on $\mathbb{R}$ converges in distribution if and only if the corresponding Lévy's metric converges (Relationship to weak toplogy (Lévy metric)).
According to ...
- 616
5
votes
1
answer
1k
views
How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric
The Lévy metric between distribution functions $F$ and $G$ is given by:
$$\rho(F,G) = \inf\left\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\right\}.$$
Another way to ...
user621937
5
votes
1
answer
781
views
Necessary and sufficient condition for weak convergence and convergence of density
Let $(\mu_n)_n$ and $\mu$ be two probability measure, having respectively density $(f_n)_n$ and $f$ for the measure $\lambda$ on $(\mathbb{R},B(\mathbb{R})).$
Prove that the following statement are ...
- 1,070
3
votes
1
answer
202
views
$P_{n}(A)\xrightarrow{n \to \infty} P(A)$ for any $P$-continuity set $A$ iff $F_{n}(x)\xrightarrow{ n \to \infty} F(x)$ for all continuity points
For a while, I have been strunggling to find why Levy's continuity Theorem would imply that:
$P_{n} \xrightarrow{\text{distribution}} P\iff \phi_{n}(t)\xrightarrow{n \to \infty}\phi(t)\; $
since Levy'...
- 998
3
votes
1
answer
1k
views
Convergence of Maximum of Cauchy Random Variables
Suppose $\{P_n\}$ and P are probability measures on the real line with corresponding distribution functions $\{F_n\}$ and $F$, respectively.
$P_n$ converges weakly to P if and only $$\lim_{n \...
- 401
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
889
views
Weak convergence: equivalence of definitions
Consider a sequence of random variables $(X_n)_{n\geq 0}$ and a random variable $X$. How to prove that the two following definitions of weak convergence are equivalent?
Def 1 $(X_n)_{n\geq 0} \...
- 200
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
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
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
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
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