# UW Probability Seminar

Time: Monday, February 8, 2016 at 2:30 pm.

Location: SAV 155

Speaker: Tvrtko Tadic (University of Zagreb and Microsoft Bing)

Title: Random reflections in a high dimensional tube

Abstract: Random reflections have been the subject of several recent papers. One of these contains results on reflections in two-dimensional and three-dimensional semi-infinite tubes. We will present results on the $n$-dimensional semi-infinite tubes, for $n\geq 3$, made of Lambertian material, i.e., material reflecting in the "most" random fashion. More precisely, the direction of a reflected light ray has the density proportional to the cosine of the angle with the normal vector. The source of light is placed far away from the end of (opening in) the tube. Using the properties of the arccosine distribution we are able to analyze the tail of the distribution of the light ray after a large number of reflections in the direction of the axis of the tube. This enables us to show that many of the results for the three-dimensional case are a special case of a more general result for the $n$-dimensional case. Further, we are able to obtain new asymptotic results on the properties of the exit distribution in dimensions $n\geq 3$ and new general results on overshoot and undershoot of a random walk. It turns out that the behavior in the two-dimensional case is different from the behavior in higher dimensions.

Joint work with Krzysztof Burdzy.

Time: Monday, February 22, 2016 at 2:30 pm.

Location: SAV 155

Speaker: David M. Mason (University of Delaware)

Title: The Breiman Conjecture

Abstract: Let $Y,Y_{1},Y_{2},\ldots$ be positive, nondegenerate, i.i.d.~$G$ random variables, and independently let $X,X_{1},X_{2},\ldots$ be i.i.d.~$F$ random variables. Breiman (1965) conjectured that if $X$ is nondegenerate, $E|X |<\infty$ and $\mathbb{T}_{n}=\sum X_{i}Y_{i}/\sum Y_{i}\rightarrow _{d}T$, where $T$ is nondegenerate, then necessarily $\overline{G}=1-G$ is regularly varying at infinity with index $% 0\leq \beta <1$, written $G\in D\left( \beta \right)$. In this talk we discuss the recent progress of P\'{e}ter Kevei and David Mason towards resolving this conjecture. We have shown that whenever for some $F\in \mathcal{F}$, in a specified class of distributions $\mathcal{F}$, $\mathbb{T% }_{n}$ converges in distribution to a nondegenerate limit then necessarily $% G\in D\left( \beta \right)$ with $0\leq \beta <1.$ The class $\mathcal{F}$ contains the distributions of nondegenerate $X$ with a finite second moment, as well as those of $X$ in the domain of attraction of a stable law with index $1<\alpha <2$. Our results will appear in the Proceedings of the AMS. We shall also discuss the limiting distributional behavior of these self-normalized sums along subsequences and their L\'evy process analogs.

Time: Monday, February 29, 2016 at 2:30 pm.

Location: SAV 155

Speaker: Brent Werness (University of Washington)

Title: TBA

Abstract: TBA

Time: Monday, March 7, 2016 at 2:30 pm.

Location: SAV 155

Speaker: James Lee (University of Washington)

Title: Embeddings, martingales, and random walks on graphs

Abstract: Consider a reversible Markov chain on a discrete metric space X; the canonical example is a random walk on a graph equipped with its path metric. We consider geometry-preserving maps from X into a linear space Y (primarily a Hilbert space). One can often use such a map (or family of maps) to associate a martingale in Y to the random walk on X, and then powerful techniques from martingale theory can be brought to bear.

The martingale lens has applications in both directions: One can study metric embeddability through random walks defined on X, and also random walks via suitably constructed embeddings. I will give examples of both, with connections to geometric group theory, harmonic functions on graphs, and the non-linear theory of Banach spaces.

This is based on joint works with Yuval Peres and various other collaborators.

Archive of previous talks

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