All Questions
34
questions
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
252
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 ...
- 5,611
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