6 votes

The drunken blind man’s walk

The expected exit time (from a ball of radius 1) is $(1+o_\delta(1))/\delta^2$, regardless of the choice of strategy (the $o_\delta(1)$ term does depend on the strategy). Indeed, write $X_n=\sum_{i=1}^...
ofer zeitouni's user avatar
4 votes
Accepted

Construction of random tempered distributions

Let me replace $\mathbb{R}_+\times \mathbb{R}^d$ by $\mathbb{R}$, the generalisation is an easy exercise. Write $\phi_n$ for the $n$th Hermite function, so that $\eta_n = \xi_{\phi_n}$ form a sequence ...
Martin Hairer's user avatar
4 votes
Accepted

Decay estimate of moment of an SDE

Since we can apply Burkholder-Davis-Gundy to control the supremum of moments, we start by removing the tail simply using $1\leq \frac{|X_{t}|}{R}$. $$ \mathbb E [ |X_t|^p 1_{\{|X_t| \ge R\}} ]\leq \...
Thomas Kojar's user avatar
  • 4,414
3 votes

If Kolmogorov continuity criterion gives the optimal Hölder regularity then does the process have all moments?

I believe that the answer is "no", here is almost a proof. Take $\tau_n$ to be an i.i.d. sequence of uniform $[0,1]$ random variables and take $Z_n$ an i.i.d. sequence of random variables ...
Martin Hairer's user avatar
1 vote

What is the expected remaining life duration of a cell in the $t\to\infty$ limit?

I don't agree. I think the exponential growth biases it towards younger individuals, that is, lots of newborns. My back of the envelope calculation is the following. Suppose the distribution of ...
mike's user avatar
  • 1,174
1 vote

Optimally betting a beta-biased coin

Let $\alpha_0$ bet the cutoff value of $\alpha$ as a function of $N$ and $\beta$ -- meaning that for $0<\alpha<\alpha_0$ it is better not to bet on the first coin flip while for $\alpha_0<\...
Claude Chaunier's user avatar
1 vote

A coupon collector-ish question

Here's a not quite complete argument: Let $t$ be the number of (not necessarily distinct) coupons you've seen so far. For a given coupon, the probability you have not seen it so far is $(1-p_i)^t$. ...
Kevin P. Costello's user avatar

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