# Coffee-Break Problem, 22/03/2021

Toss two coins, $A$ and $B$, independently of one another, with probabilities $p, q \in (0,1)$ respectively of giving heads. What is the probability

$\mathbb{P}(\text{Coin }A\text{ is heads }\implies \text{ coin }B\text{ is heads})?$

…but the coins are independent! Explain.

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# Coffee-Break Problem, 15/03/2021

Construct a complex-analytic function $f: D(0,1)\to \mathbb{C}$ on the open unit disk $D(0,1)$ such that all derivatives $f^{(n)}$ admit continuous extensions to the closed disk $\overline{D(0,1)}$, but such that $f$ cannot be analytically continued to any strictly larger domain $\Omega \supsetneq D(0,1)$.

# Coffee-Break Problem, 08/03/2021

I mentioned Sanov’s theorem in a previous post. This is a fun variant on the same argument.

Let $p_x, x\ge 0$ be a Geometric distribution on $\{0,1,....\}$ given by: $p_x=2^{-1-x}$, so that the mean is $\sum_x xp_x = 1$, and let $X_1, X_2,....X_N,....$ be independent and identical samples from this distribution. For $N\ge 1$, let $p^N$ be the empirical distribution of $X_1,....X_N$, that is, $p^N_x = N^{-1} \#\{i\le N: X_i=x\}.$

Construct explicitly probability measures $\mathbb{Q}^N \ll \mathbb{P}$ and $a \in (0, \infty)$ such that, for all $\epsilon>0$,

$\mathbb{Q}^N\left(\sum_x |p_x - p^N_x| \le \epsilon, |\sum_x x p^N_x - 2| \le \epsilon, \frac{1}{N} \log \frac{d\mathbb{Q}^N}{d\mathbb{P}}\le a\right) \to 1.$

If you know about entropy of probability measures, show also that these changes of measure have asymptotically finite entropy:

$\limsup_N \frac{1}{N}H\left(\mathbb{Q}^N \big| \mathbb{P} \right) < \infty.$

# Coffee-Break Problem, 01/03/2021

There are more counterexamples in heaven and earth than are dreamt of in your real analysis course – Hamlet, deleted scene.

Show that there exists a smooth function $f:\mathbb{R}\to\mathbb{R}$ such that, for all $x_0$, the Taylor series $\sum_{n=0}^\infty (x-x_0)^n f^{(n)}(x_0)/n!$ has zero radius of convergence.

Note that this is different from this previous problem.Why?

Bonus points for an explicit construction.

# Coffee-Break Problem, 22/02/2021

Let $\mu$ be a compactly supported probability measure on $\mathbb{R}$. For $t>0$, let $g_t(x)=\exp(-x^2/2t)/\sqrt{2\pi t}$ be the Gaussian distribution of variance $t$.

Fix a continuous function $\varphi: \mathbb{R} \to \mathbb{R}$, and for $x\in \mathbb{R}, t>0$, define a local average

$\varphi^t(x):=\left.\int_\mathbb{R} \varphi(y)g_t(y-x)\mu(dy) \right/ \int_\mathbb{R} g(z-y) \mu(dz)$

Suppose that $\mu$ has a density $f\ge 0$. Making any assumptions necessary on the regularity of $f$, show that $\varphi^t(x)$ converges to a limit $\varphi^0(x)$ for all $x\in \mathbb{R}$. Must $\varphi^0$ be continuous?

What happens if we work only with a compactly supported probability measure $\mu$?

# Coffee-Break Problem, 15/02/20

You are offered a choice between two envelopes: envelope A contains $\pounds x$, while envelope B contains $\pounds y$ with probabilitity $p\in (0,1)$, and is otherwise empty, with $py>x$. With constant average risk aversion $U(x)=-\exp(-\gamma x)$, which option do you prefer?

A prankster wishes to waste your time by making you make a large number of choices, while still making the same amount of money available for prizes. You will be given a large number $N$ choices between envelopes A, which contain $\pounds x/N$, or envelopes B, which contain $\pounds y/N$ with probability $p$, independently of each other. What is your strategy in this case? Do you complain about this prank?

Does your answer change if you are allowed to open the envelopes before making the next choice, or if you only open all of them at the end?

What is your optimal strategy if you are instead told that exactly $\lfloor pN\rfloor$ of the B envelopes contain the prize, but you do not find out about those you do not open?

# Coffee-Break Problem, 08/02/21

Consider the antiferromagnetic Ising Model on a finite graph $G=(\Lambda,E)$, where each site $v\in \Lambda$ is assigned a spin $\sigma(v)\in \{\pm 1\}$, and the probability of a given configuration is $p(\sigma) \propto \exp\left(-\beta \sum_{(v,w)\in E} \sigma(v)\sigma(w)\right)$, $\beta>0$, so that nearest neighbours are encouraged to have opposite spins.

What do typical configurations look like if $\Lambda$ is a (finite) square lattice in the limit $\beta \to \infty$? What about if $\Lambda$ is a triangular lattice?

# Coffee-Break Problem 01/02/2021

Show that the space of smooth functions $C^\infty([0,1])$ admits a complete metric $d$ such that $d(f_m, f)\to 0$ if and only if the $n^\text{th}$ derivatives $f_m^{(n)}$ converge uniformly to $f^{(n)}$ for all $n\ge 0$.

Show that the set of functions $\mathcal{A} \subset C^\infty([0,1])$ which are given by a power series about any point is meagre in the sense of Baire.

# Coffee-Break Problem, 25/01/2020

Construct a continuous local martingale $M_t, t\ge 0$ with $M_0=0$, such that $\mathbb{E} |M_t|^\alpha = \infty$ for all $\alpha>0, t>0$.

# Coffee-Break Problem, 18/01/2021

Classify all Banach spaces $X$ such that there is a surjection $f: \mathbb{R}\to X$, which is continuous when $X$ is equipped with, respectively, the norm topology or the weak topology.

What about if $X$ is no longer Banach?

This is a more high-level version of a previous post. Solving the previous problem may make this problem easier and/or less interesting.