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In short, in the perspective of preparing a new course, I am looking for examples of "concrete" (hopefully research-type) questions concerning various models in probability theory which give the opportunity to consolidate measure-theoretical tools.

More precisely, the course would be designed for students (lets say around 20 or 30) in their 5th year of studies after high school, who specialize in "applied mathematics", who have a rather good mathematical background, who have manipulated the formalism of modern probability, who have had a course in martingales and Markov chains designed for a broad audience (but invoking measure theory as less as possible). Among these students, some are interested in the more "theoretical" side, which they have yet never been taught. The goal would be twofold: consolidate the fundamentals of measure theory used in probability theory and to stimulate the curiosity of the students by showing "nice" results in probability.

Since the students are already used to probability theory, the idea of the course would be not to start by the fundamentals of measure theory, but first rather present some concrete (hopefully interesting and intriguing) questions (to motivate the students) that require tools from measure theory, and seize the opportunity to study and consolidate these theoretical aspects (for example one question for one or two lectures). For instance, one may think of:

  • The probability of having an infinite component in bond percolation on $\mathbb{Z}^d$ is $0$ or $1$: this would be the opportunity to recall Kolmogorov's $0$-$1$ law, the notion of $\sigma$-field generated by cylindrical events, etc.

  • Wigner's semi-circle law: this would be the opportunity to recall the different notions of convergence of random variables, to encounter a more complicated random variable (a random measure), to present the method of moments, to ask when a probability distribution is characterized by its moments.

Do you have other examples of "nice" and "concrete" questions in probability theory, which can be asked without too much background, and which illustrate the use of results from measure-theory?

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