A sequential particle algorithm proposed by Oudjane (2000) is studied here, which uses an adaptive random number of particles at each generation and guarantees that the particle system never dies out. This algorithm is especially useful for approximating a nonlinear (normalized) Feynman-Kac flow, in the special case where the selection functions can take the zero value, e.g. in the simulation of a rare event using an importance splitting approach. Among other results, a central limit theorem is proved by induction, based on the result of Rényi (1957) for sums of a random number of independent random variables. An alternate proof is also given, based on an original central limit theorem for triangular arrays of martingale increments spread across generations with different random sizes.
