Consider the maximisation problem

Explain why the maximum is obtained. Using the Ornstein-Uhlenbeck semigroup, show that any maximisers admit a density and enjoy a bound on .

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Consider the maximisation problem

Explain why the maximum is obtained. Using the Ornstein-Uhlenbeck semigroup, show that any maximisers admit a density and enjoy a bound on .

Let be a sequence of independent, one-dimensional Brownian motions, starting from deterministic starting points .

Suppose that and consider the (unnormalised) empirical measure . Give an example to show that need not be a locally finite measure.

Harder: prove the existence of a nonrandom such that is, with probability 1, locally finite when , while . What about the critical value ?

Fix finite sets , together with probability measures , together with a map .

I now take a random variable on and independent samples . I will now generate two new random variables by

where is nonrandom. Is it true that the entropies satisfy ? What about for `typical’ ?

Suppose that and are such that in the weak topology of , that is, for all , it holds that .

Is it true that, for any , the functions converge weakly to ? What about for almost all ? What about if we replaced by or by with ?

It is well-known that, for , all functions which are Hölder-continuous of exponent are constant. What about functions defined on the Cantor set ?

Let be a smooth, compactly supported function. Suppose we have estimates, for some ,

.

Show that these estimates imply an estimate on for some , where the upper bound depends on and one other quantity, but no other norms of . What other quantity is necessary?

Let be a Poisson random variable with mean , and fix . What are the asymptotics of

Let be Lipschitz functions. For every , consider the solution to

for a Brownian motion . Must the two-point process be a Markov process, for any ?

Let be a continuous-time Markov chain on a finite state space . Let be a function to another set; must be a Markov chain? What about the jump process which counts the jumps of ?

Let be a sequence of (Borel) probability measures on a compact metric space , which converge weakly to a point mass for some .

Suppose further that are probability density functions with respect to , i.e. that for every , and are such that for every .

Must it be true that the probability measures converge weakly to ? What about if we replace the hypothesis of weak convergence of by total variation convergence?