Is there any analogue of continuous martingale quadratic variation for the discrete case? If so, are there any theorems which characterize simple random walk using quadratic variation - similar to Levy Characterization of Brownian motion. Thanks



Quadratic variation for discrete Martingale
Certainly; if $M\_n$ is a discrete-time $L^2$ martingale, then its quadratic variation $\langle M \rangle\_n$ is the unique predictable increasing process such that $\langle M \rangle\_0 = 0$ and $M\_n^2 - \langle M \rangle\_n$ is a martingale. The existence and uniqueness follow from [Doob's decomposition](http://en.wikipedia.org/wiki/Doob_decomposition_theorem), the discrete-time (and much simpler) precursor to the Doob-Meyer decomposition.


Simple random walk is not uniquely characterized by its quadratic variation; indeed, if $X\_i$ are iid with any distribution having mean 0 and variance 1, then $M\_n = X\_1 + \dots + X\_n$ is a martingale with quadratic variation $\langle M \rangle\_n = n$. 


