For a separable metric space $M$, the Prohorov metric is a way of quantifying how far apart two probability measures on $M$ are. The metric induces the topology of weak convergence; in other words, the set of probability measures on $M$ becomes a separable metric space in its own right, and a sequence $\{P_n\}$ of probability measures converges weakly to $P$ if and only if $P_n$ converges to $P$ under the Prohorov metric. The Prohorov metric is a generalization of the Levy metric, which applies only to $S=\mathbb{R}$. Although the definition of the Prohorov metric looks tough to swallow at first, I will try to present it in a way that makes it feel as natural as possible.

Gaussian processes are a class of stochastic processes describing a wide range of phenomena. In this talk I define Gaussian process and discuss basic properties. One of the most interesting Gaussian processes is fractional Brownian motion. Fractional Brownian motion is the only centered, continuous, stationary, self similar Gaussian process. I will prove this. It finds many uses in finance, physics, biology and many other subjects. I discuss some nice properties and some not so nice properties of this process.

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In this talk, we discuss the log Sobolev inequality and its application to the concentration of measure phenomenon. I will present the proof of the inequality by L. Gross, which relies on the central limit theorem.

The Central Limit Theorem tells us that the sum of iid random variables with finite mean and variance, when centered and scaled appropriately, converges to a standard normal random variable. I will discuss generalizations of the CLT which relax the assumptions of identical distribution or independence. If time permits, we'll see what happens to the CLT when the assumption of finite variance is relaxed.

The Central Limit Theorem has several kinds of proofs. What we learned in class is by the convergence of characteristic functions. I will show you one that is proved by relative entropy. This proof is given by Andrew R. Barron in 1986. The proof itself can also be viewed as a corollary of a more general property of the relative entropy.

In 519, we briefly learned about the density function of a beta random variable with parameters $a>0$ and $b>0$, and then tried our best to forget all about it. Recall that a beta random variable takes values between 0 and 1, but the density changes dramatically depending on how you adjust the parameters. If $a$ and $b$ are positive integers, then the beta distribution with parameters $a$ and $b$ is the conditional distribution of a probability parameter $p$ for a binomial random variable, given $a-1$ successes and $b-1$ failures in $a+b-2$ attempts and given that the prior distribution for $p$ is uniform on $[0,1]$.

The Beta distribution can also be thought of in terms of Polya urns. Given an urn with $a$ green balls and $b$ red balls, suppose we pull out a ball at random, and then put it back with one more of the same color. Continue this process indefinitely, and the eventual proportion of green balls follows a Beta distribution with parameters $a$ and $b$.

The connection between these two ways of looking at the Beta distribution is surprising, as the relationship is not immediately obvious, and subtracting 1 from $a$ and $b$ in the first viewpoint but not the second seems mysterious. Rather than simply proving the relationship by calculations, I will attempt to give an explanation that feels satisfying and shows a natural connection via a third viewpoint: namely, that beta distributions are the distributions of the order statistics of multiple independent uniform random variables on $[0,1]$. Dirichlet distributions are a very natural "higher-dimensional" generalization of beta distributions, and my arguments can be generalized to these with little difficulty.

In a basic analysis course, we saw a proof of the Stone-Weierstrass Theorem using the polynomials \[P_n(x)=\int_0^1f(t)c_n(1-(x-t)^2)^ndt\] where $c_n$ satisfies $\int_0^1c_n(1-(x-t)^2)^ndt=1$. In this talk, I will give another proof of the Stone-Weierstrass Theorem using the Weak Law of Large Numbers and Bernstein polynomials given by: \[B_n(x)=\sum_{k=0}^nf(k/n)\binom{n}{k}x^k(1-x)^{n-k}\]

The archetypical infinite dimensional Gaussian measure is Wiener measure on the classical Wiener space $C_0[0,T]$. This measure is the law of Brownian motion. In this talk, we discuss how to generalize Gaussian measures on $\mathbb{R}^n$ to Gaussian measures on general Banach or even locally convex topological vector spaces. One of the most important results in infinite dimensional Gaussian measure theory is Fernique's theorem. We discuss Fernique's theorem and may or may not offer a proof, as is true in any talk.

Percolation is one of the simplest probabilistic models that exhibits what is referred to as a critical phenomenon. In this talk we’ll take a brief exploration to the field of percolation theory. We will start by looking at the original motivation behind the field, the different variations on the model, and important results in the field. In particular, we will sketch a proof of the existence of the critical probability $p_c(d)$ in different dimensions. We will conclude the talk by discussing other known results and open problems in the field.

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During a conversation with Daniel Slonim, he mentioned how much he cares about rates of convergence. Due to his enthusiasm for rates of convergence I had to indulge him. Stein's method is a technique for determining if a measure is close to a Gaussian. One of the big results is Berry Esseen theorem which is a quantitative version of central limit theorem. In this talk I will discuss Stein's heuristic, state and solve Stein's equation and use this to prove bounds on TV distance.

I shamelessly take this opportunity to practice for my advanced topics exam. As the name "random walks in random environments" suggests, the model contains two levels of randomness. We first consider random walks on the integers. An environment $\omega$ is an assignment of a "stepping right" probability $\omega_x$ to each integer $x$. The first level of randomness is in how the probabilities $\omega_x$ are assigned, which we assume occurs in an iid way. Once an environment is fixed, we consider a random walk on the integers where at every point $x$, the probability of stepping to the right is $\omega_x$ and the probability of stepping to the left is $1-\omega_x$. I will give a characterization of directional transience (i.e. whether the walk wanders off to infinity or negative infinity or is recurrent), and perhaps a law of large numbers with an explicit formula for limiting velocity.

My primary interest is in higher dimensional models, where the walk is on $\mathbb{Z}^d$ rather than $\mathbb{Z}$, and the transition probabilities are still chosen in an iid random way. In this setting, there is no explicit formula for limiting velocity or explicit characterization of directional recurrence and transience. In fact, the situation is even worse than this, and I will explain why. I will, however, give a couple of results about directional transience and limiting velocity that are known. Time permitting, I will also sketch a proof of a functional central limit theorem under certain conditions.