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
.
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?