  The Kalman filter combines forecasts and new observations to obtain an
estimation which is optimal in the sense of a minimum average quadratic error.
The Kalman filter has two main restrictions: (i) the dynamical system is
assumed linear and (ii) forecasting errors and observational noises are taken
Gaussian. Here, we offer an important generalization to the case where errors
and noises have heavy tail distributions such as power laws and L\'evy laws.
The main tool needed to solve this ``Kalman-L\'evy'' filter is the
``tail-covariance'' matrix which generalizes the covariance matrix in the case
where it is mathematically ill-defined (i.e. for power law tail exponents $\mu
\leq 2$). We present the general solution and discuss its properties on
pedagogical examples. The standard Kalman-Gaussian filter is recovered for the
case $\mu = 2$. The optimal Kalman-L\'evy filter is found to deviate
substantially fro the standard Kalman-Gaussian filter as $\mu$ deviates from 2.
As $\mu$ decreases, novel observations are assimilated with less and less
weight as a small exponent $\mu$ implies large errors with significant
probabilities. In terms of implementation, the price-to-pay associated with the
presence of heavy tail noise distributions is that the standard linear
formalism valid for the Gaussian case is transformed into a nonlinear matrice
equation for the Kalman-L\'evy filter. Direct numerical experiments in the
univariate case confirms our theoretical predictions.
