Analysis of the Second Moment of the LT Decoder

We analyze the second moment of the ripple size during the LT decoding process and prove that the standard deviation of the ripple size for an LT-code with length $k$ is of the order of $\sqrt k.$ Together with a result by Karp et. al stating that the expectation of the ripple size is of the order of $k$ [3], this gives bounds on the error probability of the LT decoder. We also give an analytic expression for the variance of the ripple size up to terms of constant order, and refine the expression in [3] for the expectation of the ripple size up to terms of the order of $1/k$, thus providing a first step towards an analytic finite-length analysis of LT decoding.