Blogposts

We extend the theory of Tukey depth to the setting of absolutely regular (\(\beta\)-mixing) stochastic processes. In particular, we generalize Massé's weak convergence result for the Tukey depth process to dependent observations and establish convergence rates expressed in terms of the mixing coefficients. An application extends the theory developed by Massé on depth weighted estimators. First, we show that the assumptions on the weight can be significantly weakened, and secondly we show that the observations can be taken as weakly dependent. A simulation study demonstrates the advantages of depth weighted estimators under heavy tailed innovations.

An introductory and an in-depth talk from each of Susanne Ditlevsen and Johannes Ruf, running 10:00–16:15 with breaks, followed by drinks. Talk titles TBA.

STAR Lunteren is the main annual meeting for probability and statistics researchers in the Netherlands, a few days at a conference center in Lunteren that most Dutch PhD students in the field end up at every year. I'm going as a poster presenter this time; I'll add the poster itself once it's ready.

A week-long virtual market-making competition: participants write a trading bot in Python that competes against Akuna's own models and other entrants on a simulated exchange. It's a practical test of the same market-making problem I work on professionally — quoting, managing inventory, and reacting to order flow — but under a fixed, competitive ruleset rather than production constraints.

Gave a talk titled “Maximal and concentration inequalities for mixing empirical measures and their application.” Empirical process theory gives sharp control of how an empirical measure fluctuates around its target under independence; the talk covers how much of that control survives once the independence assumption is dropped in favor of mixing conditions, and what the resulting inequalities are useful for in practice.

A talk on a new construction showing that, once three colors are allowed, the van der Waerden numbers grow faster than any exponential function of the progression length, resolving a question that had been open for about a hundred years about their true rate of growth, and settling several conjectures that had proposed the opposite.

The van der Waerden number

For positive integers \(k\) and \(r\), the van der Waerden number \(w(k;r)\) is the smallest positive integer \(N\) such that every coloring of the integers from one to \(N\) using \(r\) colors contains a monochromatic arithmetic progression of length \(k\). The theorem of van der Waerden guarantees that such an \(N\) always exists; the difficulty, open for a century, is determining how quickly \(w(k;r)\) grows as \(k\) grows.

Erdős conjectured that \(w(k;2)\) grows faster than any exponential function of \(k\), that is, \[\limsup_{k \to \infty} w(k;2)^{1/k} = \infty,\] while an opposing conjecture proposed instead that \(w(k;r)^{1/k}\) converges to \(r\) for every fixed number of colors \(r\).

The result

The talk establishes that, for three or more colors, the opposing conjecture is false. The van der Waerden numbers grow super-exponentially in \(k\) whenever \(r \ge 3\). Writing \(\log^{*} k\) for the iterated logarithm of \(k\) (the number of times the logarithm must be applied to \(k\) before the result is at most one), the precise bound proved is that, for \(k\) sufficiently large, \[w(k;3) > 2^{k (\log^{*}k)/4}.\]

The construction underlying this bound is a randomized “shifted product” procedure. A very dense, arithmetic-progression-free subset of a large cyclic group is built probabilistically, then combined with smaller constructions of the same kind to produce successively larger three-colorings free of monochromatic \(k\)-term progressions, iterated roughly \((\log^{*}k)/2\) times. The same circle of ideas also yields a new lower bound on the canonical van der Waerden numbers \(H(k)\), resolving a related open problem of Erdős and Graham by showing \[H(k) \ge k^{(1-o(1))k\log k}.\]

A talk on joint work with Amol Aggarwal showing that the Toda lattice, a classical system of interacting particles on the real line, has current and particle fluctuations that converge, after diffusive rescaling, to an explicit Gaussian process, placing this integrable system in a different universality class from the non-Gaussian fluctuations expected of comparable chaotic particle systems.

The Toda lattice at thermal equilibrium

The Toda lattice is a Hamiltonian system of particles indexed by the integers, with positions \(q_i(t)\) and momenta \(p_i(t)\) evolving under the equations of motion \[\partial_t q_i(t) = p_i(t), \qquad \partial_t p_i(t) = e^{q_{i-1}(t) - q_i(t)} - e^{q_i(t) - q_{i+1}(t)}.\] Because it possesses infinitely many conserved quantities, the Toda lattice is a classical example of an integrable system, in contrast to generic, chaotic many-body Hamiltonian systems. The talk studies it under thermal equilibrium, the natural random initial condition in which momenta and position increments are sampled independently from explicit Gaussian and gamma distributions.

Diffusive Gaussian fluctuations

For chaotic interacting particle systems, physical predictions and rigorous results for related stochastic models place space-time current fluctuations after a long time \(T\) at the \(T^{1/3}\) scale, converging to a non-Gaussian limit belonging to the Kardar–Parisi–Zhang universality class. For the integrable Toda lattice, the talk instead establishes that these fluctuations sit at the larger \(T^{1/2}\) scale and converge to an explicit Gaussian process. As one consequence, the trajectory of a single particle, suitably rescaled, converges to a Brownian motion: \[T^{-1/2} \cdot q_0(T\tau) \longrightarrow \mathcal{B}(\tau), \qquad T \to \infty.\]

The proof views the lattice as a dense collection of interacting “quasi-particles,” each carrying a conserved spectral parameter and a location that moves at an explicit effective velocity between collisions, and shows that the joint fluctuations of all quasi-particles converge to a Gaussian process termed a dressed Lévy–Chentsov field.

Presented a poster on joint work with Dr. A.M. Dürre studying the Tukey depth under short-range dependence: how the classical notion of statistical depth, usually studied for independent samples, behaves once the underlying data has short-range temporal dependence instead.

Talks covered functional estimation of option pricing models by Evgenii Vladimirov, valuation of interest rate derivatives on arithmetic averages of risk-free rates by Arco de Kort, and measuring financial resilience using backward stochastic differential equations by Matteo Ferrari.

The fourth Finance Research Day organized by DIAM at TU Delft: a one-day mix of academic talks and industry perspectives on quantitative finance, aimed at getting researchers and practitioners in the same room for once.

Presented a poster on maximal inequalities and concentration of measure with absolute regularity: how far the classical toolkit for bounding suprema of stochastic processes extends once the underlying process satisfies absolute regularity (beta-mixing) rather than independence.