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The optimal fllter … = f…t; t ‚ 0g for a general observation model is approximated by a probability measure valued process …n = f…n ; t ‚ 0g. The process … n is the empirical measure of a system of weighted particles that at time 0 consists of n particles. The particles branch at equally spaced time instances jn i2fi where j = 1; 2; ::: and 0 < fi < 1. We prove the convergence of the process … n to … and derive sharp upper bounds for the mean square error. We also prove a central limit theorem to characterize the convergence rate of the approximate fllter. A similar result is obtained for the unweighted, unnormalized version introduced in (8). As a corollary, we show that fi = 1 is the optimal exponent for that version. The approximation of the optimal nonlinear fllter by means of particle approx- imations has been studied extensively in last ten years (see, for example, (5), (7), (10), (12) and the references therein). The use of particle approximations stems from the fact that the unnormalized fllter can be approximated by a weighted particle system. Since the weights have variances which grow exponentially fast, the particle system needs to be corrected after small time steps to control the error. At each time step, the particles will be replaced by a random number of \ofisprings". The expected number of ofisprings is the weight of the corresponding particle decided according to its path during the period prior to that time step. In the following, we will work within a very general flltering framework. Namely, we will assume that the observation process takes values in a space of measures (rather than the usual k-dimensional Euclidean space). In particular, the observa- tion process can be given by a random measure in space and time. Moreover, we will allow the observation and signal noises to be correlated. Let us now introduce the flltering model in more detail. Let (›;F; (Ft) t‚0 ;P0) be a flltered probability space and (H; h¢;¢i H ), respec- tively i H1; h¢;¢i H1 ¢ be two real separable Hilbert spaces such that H µ H1 and the injection from H to H1 is a Hilbert-Schmidt operator. On (›;F; (Ft) t‚0 ;P0) we deflne an Ft-adaptedd-dimensional Brownian motion B = fBt;Ft; t ‚ 0g and an Ft-adapted H-cylindrical Brownian motion (H-c.B.m.) W = fWt;Ft; t ‚ 0g

A central limit type theorem for a class of particle filters