All Questions
8
questions with no upvoted or accepted answers
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
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 $$\...
- 155
2
votes
0
answers
59
views
Question on the weak convergence of measures implying convergence of integrals over some boundaryless set
This is a setting from Ken Iti Sato's Levy Processes.
Define $$g(z,x) = e^{i \langle z,x \rangle} -1 - i\langle z,x \rangle c(x)$$
where $c(x) = 1+o(|x|)$ as $|x|\to 0$ and $c(x)$ is some bounded ...
- 5,611
2
votes
0
answers
394
views
Convergence of Probability Measures and Respective Distribution Functions
Suppose $\{P_n\}$ and P are probability measures on the real line with corresponding distribution functions $\{F_n\}$ and $F$, respectively.
Prove that $P_n$ converges weakly to P if and only $$\lim_{...
- 401
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
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
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