Coffee-Break Problem, 02/11/20

Consider the following model for fruit fly populations: every fruit fly dies after a time T \sim \text{Exponential}(\lambda) , and is then replaced by two identical fruit flies. These fruit flies then reproduce in the same way, identically to their parent (and all other flies).

Fruit flies are small, but make up for this smallness by a large population. Writing X_t for the number of fruit flies at time t , let us consider a regime in which X_t=\mathcal{O}(N) – so that Y_t=X_t/N , representing the total mass of flies, is a macroscopic object.

Suppose that the initial number of flies X_0 is distributed according to a \text{Poisson}(N\delta) distribution, for some small \delta>0 , and we want to know the probability that X_t/N >x for some fixed t>0, x>0 . Show that, no matter how small \delta >0 is, there is a certain (exponentially small) probability of this happening:

\liminf_{\delta\to 0}\left[\liminf_{N\to \infty}\frac{1}{N} \log \mathbb{P}\left(X_t /N >x\right)\right]>-\infty.

What happens if we take the limits N\to \infty,  \delta \to 0 in the other order?

Coffee-Break Problem, 12/10/20

Suppose that I am running a speed-meeting session, where I ‘randomly’ assign 2n participants into pairs. This is repeated over a number of rounds, hoping that each participant meets several others.

On average, how many people are assigned the same partner in two consecutive rounds? If I run m rounds, how many new people (on average) does each participant meet?

Coffee-Break Problem, 05/10/20

Let (\Omega, \mathcal{F}, \mathbb{P}) be the the product probability space given by (\{0,1\}, \{\emptyset,\{0\},\{1\},\{0,1\}\}, \frac{1}{2}\delta_0+ \frac{1}{2}\delta_1)^{\otimes \alpha} , for an uncountable ordinal \alpha , and let (\mathcal{F}_t)_{0\le t\le 1} be a filtration with \mathcal{F}_1=\mathcal{F} .

On this probability space, let \mathcal{M} be the space of all L^2 -bounded martingales (M_t)_{0\le t\le 1} , equipped with the distance d(M,N):=(\mathbb{E}|M_1-N_1|^2)^{1/2} . Define also \mathcal{R} as martingales with some Hölder-regularity:

\mathcal{R}=\big\{M\in \mathcal{M}: \text{ for some nonempty open interval }I\subset [0,1]\text{ and }\beta>0, \mathbb{P}(M \text{ is }\beta\text{-H\"older continuous on }I)>0\big\}.

Show that \mathcal{R} \subset \mathcal{M} is meagre in the sense of Baire.