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UID:8545@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20250115T103000
DTEND;TZID=Europe/Paris:20250115T120000
DTSTAMP:20250113T100755Z
URL:https://www.i2m.univ-amu.fr/evenements/skorokhod-spaces-and-convergenc
 e-of-measure-valued-processes/
SUMMARY:Virgile Brodu (Université de Lorraine): Skorokhod spaces and conve
 rgence of measure-valued processes
DESCRIPTION:Virgile Brodu: The most famous example of convergence of stocha
 stic 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 re
 scaled 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].\\]\nW
 e consider $(Y^N)_{N \\in \\mathbb N^*}$ as a random sequence in $\\mathca
 l C([0\, 1]\, \\mathbb R)$ endowed with the topology of uniform convergenc
 e. Then\, the so-called Donsker’s theorem states that $(Y^N)_{N \\in \\m
 athbb N^*}$ 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 theo
 rem and the Central Limit Theorem.\nBut now\, what happens if we want to s
 tudy the convergence of discontinuous real-valued stochastic processes\, w
 hich is often the case for modelling purposes? For example\, think of trac
 king the evolution of the population size of living species\, where deaths
  are instantaneous negative jumps... In 1956\, Skorokhod proposed a topolo
 gy on the space of discontinuous functions\, which is predominant today. T
 he first aim of this talk is to explain the simple and intuitive ideas und
 erlying the construction of Skorokhod to facilitate its understanding\, wi
 thout going in the depth of technical proofs. In a second part\, we will i
 ntroduce measure-valued processes\, with biological motivations\, and expl
 ain how the Skorokhod construction can be generalized to more complex spac
 es such as these measure spaces.
CATEGORIES:Séminaire,Doctorant⋅es de l'I2M
LOCATION:Saint-Charles - FRUMAM  (2ème étage)\, 3 Place Victor Hugo\, Mar
 seille\, 13003\, France
X-APPLE-STRUCTURED-LOCATION;VALUE=URI;X-ADDRESS=3 Place Victor Hugo\, Marse
 ille\, 13003\, France;X-APPLE-RADIUS=100;X-TITLE=Saint-Charles - FRUMAM  (
 2ème étage):geo:0,0
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