  Products of random $2\times 2$ matrices exhibit Gaussian fluctuations around
almost surely convergent Lyapunov exponents. In this paper, the distribution of
the random matrices is supported by a small neighborhood of order $\lambda>0$
of the identity matrix. The Lyapunov exponent and the variance of the Gaussian
fluctuations are calculated perturbatively in $\lambda$ and this requires a
detailed analysis of the associated random dynamical system on the unit circle
and its invariant measure. The result applies to anomalies and band edges of
one-dimensional random Schr\"odinger operators.
