Probability – Page 4 – Applied Probability Notes

Markov Chains: a functional view

Laplacian; Adjoints; Harmonic fn; Green’s fn; Forward Eqn; Backward Eqn.
Markov Chains and Martingales; Green’s Functions and occupancy; Potential functions; time-reversal and adjoints.

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Spitzer’s Lyapunov Ergodicity

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

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A Mean Field Limit

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

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Cross Entropy Method

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

$l = {\mathbb P} ( S(X) \geq \gamma ).$

Here $X$ is a random variable whose distribution is known and belongs to a parametrized family of densities $f( , v)$ . Further $S(X)$ is often a solution to an optimization problem.

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Sanov’s Theorem

Sanov’s asks how likely is it that the empirical distribution some IIDRV’s is far from the distribution. And shows that the relative entropy determines the likelihood of being far.

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Entropy and Boltzmann’s Distribution

Entropy and Relative Entropy occur sufficiently often in these notes to justify a (somewhat) self-contained section. We cover the discrete case which is the most intuitive.

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Ito’s Formula: a heuristic derivation

A heuristic look at the stochastic integral.
heuristic derivation of Itô’s formula.

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