16 votes
Accepted

Expected length of longest stick in a stick snapping process

It seems that the length of the longest stick is of order $n^{2\sqrt{2}-3} = n^{-0.171\ldots}$ as $n\to\infty$. This follows from a discrete-time analogue of the homogeneous fragmentation process, see ...
  • 3,305
6 votes

Expected length of longest stick in a stick snapping process

UPDATE#2. Just to illustrate how the complexity of exact expressions grows, these are the first three: \begin{split} n=1:~~~& \frac{3}{4} = 0.75\\ n=2:~~~& \frac38 + \log\frac43 = 0....
5 votes
Accepted

Range of $a$ such that $w \leftarrow w-a x \langle w, x \rangle$ converges almost surely?

As you say, the threshold is around $a=0.937087$. The exact condition for convergence is $$ \int_{0}^{2\pi}\int_0^\infty\log(1-2ar^2\cos^2\theta+a^2r^4\cos^2\theta)re^{-r^2/2}\,dr\,d\theta<0. $$ I ...
  • 21.7k
4 votes
Accepted

How fast does this Gaussian random walk move away from the origin?

$\newcommand\1{\mathbf1}\newcommand\R{\mathbb R}\newcommand\Si{\Sigma}\newcommand{\si}{\sigma}\newcommand{\Ga}{\Gamma}$Letting $$y_i:=g(z_i)z_i=z_iz_i^\top\1,$$ where $\1:=[1,\dots,1]^\top\in\R^d$, we ...
3 votes

Expected length of longest stick in a stick snapping process

Here's a few empirical results that might be helpful, particularly in lending credence towards some of the theories suggested in other posts. I ran 10,000 simulations, and for each one, broke the ...
  • 3,481
3 votes

Particularities about the honeycomb lattice for the computation of connectivity constant

It's worth mentioning that there is another lattice for which the precise value of the connective constant has been established. In the paper "Self-avoiding walks and trails on the $3.12^2$ ...
  • 2,166
3 votes

Expected length of longest stick in a stick snapping process

This is intermediate between an answer, and a long comment on Max Alekseyev's post. As discussed in the comments below his answer, there is a flaw in his post, so that it gives a lower bound. But ...
3 votes

Maximizing expectation of gaussian process over covariance matrix with fixed trace

The maximizer over $\Sigma$'s with nonnegative entries is $\Sigma=I_n$. This follows from Slepian's lemma and the formula $$EZ=\int_0^\infty dz\,P(Z>z)-\int_{-\infty}^0 dz\,P(Z<z), \tag{1}\...
3 votes
Accepted

Lebesgue differentiation theorem at a stopping time

Yes. Let $h \in (0,1)$, $Q_h = \frac{1}{W_{\sigma + h} - W_{(\sigma - h) \vee 0}} \int_{(\sigma - h)\vee 0}^{\sigma + h} (H_s - H_{(\sigma - h) \vee 0}) \, dW_s$ and set $M_h = \sqrt{-\log(h)}$. ...
2 votes
Accepted

What happens in the difference rate between these two versions of ballot theorem?

$\newcommand{\de}{\delta}\newcommand{\vpi}{\varphi}\newcommand\ep\varepsilon$The paper linked in your post contains a reference to a reference to an apparent proof, which I hope results in an overall ...
2 votes
Accepted

What is a tensor product of random variables?

In this context, I believe the tensor product on random variables is nothing other than the tensor product over the values of the RVs. (In other words, if $\Omega$ is a sample space and $X : \Omega \...
1 vote

Maximizing expectation of gaussian process over covariance matrix with fixed trace

I assume all random variables are centered. It is clear that $f_1$ and $f_2$ are minimized when $\Sigma_{ij}=1$, i.e., the Gaussian vector is distributed as $(X,X,\dots,X)$ with $X \sim N(0,1)$. The ...

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