Skorokhod spaces and convergence of measure-valued processes
Virgile Brodu
Université de Lorraine
Date(s) : 15/01/2025 iCal
10h30 - 12h00
The most famous example of convergence of stochastic processes is the following. Consider a simple random walk $(S_n)_{n \in \mathbb N}$ on $\mathbb Z$ starting at $S_0 = 0$, and define the rescaled and interpolated continuous process \[Y^N_t := \frac{1}{\sqrt{N}} (S_{\lfloor Nt \rfloor} + (Nt – \lfloor Nt\rfloor )(S_{\lfloor Nt \rfloor +1} – S_{\lfloor N t \rfloor} ), \quad t \in [0, 1].\]
We consider $(Y^N)_{N \in \mathbb N^*}$ as a random sequence in $\mathcal C([0, 1], \mathbb R)$ endowed with the topology of uniform convergence. Then, the so-called Donsker’s theorem states that $(Y^N)_{N \in \mathbb N^*}$ converges in law to a standard Brownian motion $B$ on $[0, 1]$. To prove such a convergence of continuous processes, probabilists rely on a strong machinery, using Prokhorov’s Theorem, Arzelà-Ascoli theorem and the Central Limit Theorem.
But now, what happens if we want to study the convergence of discontinuous real-valued stochastic processes, which is often the case for modelling purposes? For example, think of tracking the evolution of the population size of living species, where deaths are instantaneous negative jumps… In 1956, Skorokhod proposed a topology on the space of discontinuous functions, which is predominant today. The first aim of this talk is to explain the simple and intuitive ideas underlying the construction of Skorokhod to facilitate its understanding, without going in the depth of technical proofs. In a second part, we will introduce measure-valued processes, with biological motivations, and explain how the Skorokhod construction can be generalized to more complex spaces such as these measure spaces.
Emplacement
Saint-Charles - FRUMAM (2ème étage)
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