We show that relative entropy decreases for continuous time Markov chains.

## Sums and Limits of Coin Throws

We explain why certain distributions arise naturally as the limit of coin throws.

- Bernoulli, Binomial Distributions, Geometric Distributions.
- Binomial to Poisson Distribution; Geometric to Exponential; Binomial to Normal.

## A Mean Field Limit

We consider a system consisting of interacting objects. As we let the number of objects increase, we can characterize the limiting behaviour of the system.

## Lyapunov functions

Lyapunov functions are an extremely convenient device for proving that a dynamical system converges.

## Revenue Equivalence

For a number of differing auction settings, we consider the sale of a single item amongst fixed number of auction participants. It is interesting that under a certain game-theoretic construction all these auctions can be seen to be equivalent.

## Cross Entropy Method

In the *Cross Entropy Method*, we wish to estimate the likelihood

Here is a random variable whose distribution is known and belongs to a parametrized family of densities . Further is often a solution to an optimization problem.

## Online Convex Optimization

We consider the setting of sequentially optimizing the average of a sequence of functions, so called *online convex optimization*.