  We investigate the use of possibly the simplest scheme for the
parallelisation of the standard particle filter, that consists in splitting the
computational budget into $M$ fully independent particle filters with $N$
particles each, and then obtaining the desired estimators by averaging over the
$M$ independent outcomes of the filters. This approach minimises the
parallelisation overhead yet displays highly desirable theoretical properties.
Under very mild assumptions, we analyse the mean square error (MSE) of the
estimators of 1-dimensional statistics of the optimal filtering distribution
and show explicitly the effect of parallelisation scheme on the convergence
rate. Specifically, we study the decomposition of the MSE into variance and
bias components, to show that the former decays as $\frac{1}{MN}$, i.e.,
linearly with the total number of particles, while the latter converges towards
$0$ as $\frac{1}{N^2}$. Parallelisation, therefore, has the obvious advantage
of dividing the running times while preserving the (asymptotic) performance of
the particle filter. Following this lead, we propose a time-error index to
compare schemes with different degrees of parallelisation. Finally, we provide
two numerical examples. The first one deals with the tracking of a Lorenz 63
chaotic system with dynamical noise and partial (noisy) observations, while the
second example involves a dynamical network of modified FitzHugh-Nagumo (FH-N)
stochastic nodes. The latter is a large dimensional system ($\approx3,000$
state variables in our computer experiments) designed to numerically reproduce
typical electrical phenomena observed in the atria of the human heart. In both
examples, we show how the proposed parallelisation scheme attains the same
approximation accuracy as a centralised particle filter with only a small
fraction of the running time, using a standard multicore computer.
