The celebrated fourth moment theorem of Nualart and Peccati (2005)
states that, for homogeneous Wiener chaoses of a fixed order, the
convergence of the second moments to 1 and of the fourth moments to 3
implies the convergence of chaoses in distribution to the standard
normal law. On the other hand, Rosinski and Samorodnitsky (1999)
observed that homogeneous Wiener chaoses are independent if and only if
their squares are uncorrelated. 

In this talk we relate both results and show that the fourth moment
theorem follows from an analogous criterion for the asymptotic
independence of Wiener chaoses. Furthermore, we derive a
multidimensional version of the fourth moment theorem, applicable in
the study of stochastic processes, give new bounds on the rate of
convergence, and show other related results involving Gaussian and non
Gaussian limits. Applications to the limit theory of short and long
range dependent stationary Gaussian time series will also be discussed.
If time permits, an extension to a non-Gaussian discrete chaos will be
mentioned.  This talk is based on a joint work with Ivan Nourdin.

Two-dimensional lattice models at continuous phase
transitions are expected (and sometimes proven) to exhibit conformal
symmetry in the scaling limit. As a result, their scaling limits can
be described (rigorously or conjecturally) by Conformal Field Theory
and Schramm-Loewner Evolution. Very interesting algebraic, geometric
and probabilistic structures emerge, that yield in particular exact
formulae.

I will mostly discuss the case of the Ising model, which is exactly
solvable on the lattice level, a feature that allows one to pass to
the scaling limit, to prove conformal symmetry and to connect the
model with continuous theories. I will in particular discuss how the
scaling limits of the fields and curves of the model can be
identified, and how one can understand some of the algebraic
structures behind those.

The results rely mostly on complex analytic and probabilistic
techniques.

Based on joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, K.
Izyurov, F. Johansson Viklund, A. Kemppainen, K. Kytölä, D.H. Phong
and S. Smirnov

[NOTE DIFFERENT TIME]

First question: For a non-decreasing, concave utility function U, is
E[U(X)] >= E[U(Y)] if Y is a Martingale dilation of X? The answer is
certainly positive, as U(X) is a supermartingale.
Second question: For a non-decreasing, concave utility function U, is
max_{0 <= a <= 1}E[U(aX+(1-a)*b)] >= max_{0 <= a <= 1}E[U(aY+(1-a)*b)]
for all b, if Y is a Martingale dilation of X? The answer is positive
as well, as can be ascertained after a little thinking: the best “a”
for Y works better for X because of the first question, and the best
“a” for X is even better.
Third question: Is argmax_{0 <= a <= 1}E[U(aX+(1-a)*b)] >= argmax_{0 <=
a <= 1}E[U(aY+(1-a)*b)] for all b, if Y is a Martingale dilation of X?
It of course makes sense, but the answer is not necessarily positive.
It becomes positive if x U’(x) is concave and U’(x) is convex.
Rothschild & Stiglitz 1971.
Martingale dilation is equivalent to Skorokhod embeddability in
Brownian Motion in later time.
 
Every mean-zero distribution can be Skorokhod-embedded in Brownian
Motion by the Azema-Yor stopping time  but not every Martingale
dilation is embedded by the Azema-Yor stopping time in later time.
 
Every mean-zero distribution can be Skorokhod-embedded in the Azema
Martingale but not every Martingale dilation is embeddable in the Azema
Martingale in later time.
 
Embeddability in Brownian Motion in monotone time by the Azema-Yor
stopping time and embeddability in monotone time in the Azema
Martingale are two different concepts, that will be crisply identified
in the talk.
 
For the third question above, of whether Martingale dilation controls
not only the desirability of portfolios but also the demand for the
risky asset within the portfolio, the answer will be shown to be
positive if
 
(i)		   the concave function U has x U’(x) concave and the
dilation is embeddable in monotone time in the Azema Martingale.
or if
(ii)		   the concave function U has U’ convex and the
dilation is embeddable in Brownian Motion in monotone time by the
Azema-Yor stopping time.
 
The Azema Martingale, with trajectories looking like a fishbone,
getting continuously and monotonically away from zero and then suddenly
jumping back to zero, can be well understood as the conditional
expectation of Brownian Motion B given the filtration generated by
sign(B).