All Questions
16 questions
1
vote
1
answer
160
views
Estimates of product of eigenvalues gaps for Wigner matrices
Let $W_n$ be an $n\times n$ Wigner matrix$^{1}$, and let $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$ be the eigenvalues of $\frac{W_n}{\sqrt{n}}$.
My question. For any fixed $i\in\{1,\dots,n\}$, ...
- 2,712
4
votes
0
answers
179
views
As increasingly higher degree terms are added to a "random" polynomial, how fast do the roots approach the unit circle?
As increasingly higher degree terms are added to a "random" polynomial, the roots of a polynomial can be proven to approach the unit circle. For example, see the MathOverflow question Why ...
4
votes
1
answer
364
views
Reference for multivariate generalised CLT
I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 380
1
vote
1
answer
190
views
Hitting time estimates
In a number of different contexts, I have wanted to estimate hitting times for a monotonic process $(T_n)$ taking values in the reals (or sometimes a process $(T_n,X_n)$ taking values in $\mathbb R^2$ ...
- 23.2k
1
vote
0
answers
190
views
Rate of convergence for difference between conditional and marginal probability
Suppose $X\sim \text{Bin}(2n,p)$ and $X_1,X_2\sim\text{Bin}(n,p)$ are independent, with $X_1+X_2=X$. I'm interested in the rate of convergence for the absolute difference
$$
\left\vert P(X>c|X_1\...
- 59
3
votes
2
answers
100
views
Left tail of convex combinations of $\chi_1^2$
Suppose $a_1,...,a_n\geq0, \sum_{i=1}^na_i=1$ and $Z_1,...,Z_n$ are i.i.d. standard normal, what is a sharp upper bound of the following probability as $\delta\to0$ and what is the order?
$$\mathbb{P}(...
- 1,495
7
votes
3
answers
790
views
Expected cardinality of a randomly chosen element of the family of subsets of $\{1,\ldots,n\}$ with at most $k$-elements
Assume that $1\le k \le n$ and let $\mathscr{Z}$ be the family of all subsets of $\{1,\ldots,n\}$ with at most $k$ elements. Pick a random element $X$ of $\mathscr{Z}$ (we consider the probablity ...
- 1,658
1
vote
1
answer
276
views
What is the expected number of missing random integers?
Consider $n$ numbers randomly generated by independent generators that can produce integers from $0$ to $n$. How many of these integers will be missing on average for large $n$? If $p_{k,n}$ is the ...
- 1,731
12
votes
1
answer
883
views
The dance marathon problem
In his book, "The Strange Logic of Random Graphs", Joel Spencer describes the "Dance Marathon" problem:
Imagine $n$ couples at a Dance Marathon. Each dance each couple remains ...
- 3,979
0
votes
1
answer
660
views
Tail bound for maximum of independent (but not identical) binomial random variables
This post derived the tail bound for the maximum of independent and identically distributed binomial r.v.'s based on normal approximation. Is there a similar result in the literature for finding the ...
- 59
11
votes
1
answer
1k
views
What are some of the surprising results of finite sample statistical estimation?
I'm trying to familiarize myself with the latest results in finite sample statistics. It seems to me that these results can be classified into two categories:
Unsurprising results confirm that the ...
- 395
21
votes
3
answers
2k
views
Probability of two vectors lying in the same orthant
Let $S^{d-1} = \{x \in \mathbb R^d: \|x\| = 1\}$ denote the unit sphere in $\mathbb R^d$. Let $v$, $w$ be drawn uniformly at random from $S^{d-1}$, conditioned on their inner product being equal to $\...
- 733
4
votes
2
answers
542
views
References on law of large numbers, CLT and iterated logarithm laws
Having access to those references, accumulating many results in one domain is always a bless, like Feller's book in probability, Dembo-Zeitouni's large deviation, Grimmett's percolation and recent ...
- 752
3
votes
2
answers
4k
views
Asymptotics of the maximum of binomial random variables
Let $B_i(n,1/2)$ be independent identically distributed binomial random variables. I am interested in the asymptotic growth of the maximum of $n$ such random variables. In https://math.stackexchange....
user32786
5
votes
0
answers
215
views
Asymptotics of a Splitting Process
Consider $p(n)$ defined recursively by $p(1)=1$ and
$\displaystyle p(n)=\frac{1}{(n-1)^n}\sum_{i=1}^{n-1}\left\{\sum_{j=i}^{n-1}(-1)^{j-i}{n \choose j}{j\choose i}(n-j)^j(n-j-1)^{n-j}\right\}p(i)$.
...
- 667
5
votes
0
answers
227
views
Number of times lead changes in a multi-candidate election (reference-request)
In a two candidate election where votes are distributed uniformly at random between the candidates, the probability that the lead changes when tallying the $i$-th vote is the same as the probability ...
- 51