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### 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

### 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....
• 26.8k
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### 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
Accepted

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

• 82.3k

### 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

### 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

### 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 ...

### 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}\...
• 82.3k
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### 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 $M_h = \sqrt{-\log(h)}$. ...
• 5,570
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### What happens in the difference rate between these two versions of ballot theorem?

• 82.3k