On the Superposition of Renewal Processes

Suppose that there are a number of independent sources at each of which events occur from time to time. The intervals between successive events at any one source are assumed to be independent random veriables all with the same distribution, so that each source constitutes a renewal process of a familiar type. The outputs of the sources are combined into one pooled output. Statistical properties of the pooled output are investigated and the results applied to a problem in neurophysiology. This work was suggested by problems in neuro-physiology, although there are very probably other applications. Suppose that a number of neurons independently send discrete pulses to a common central nerve cell. We shall investigate the relation between the statistical properties of the sequence of impulses from an individual neuron and the corresponding properties of the combined sequence of pulses at the central cell. A very similar problem arose in the recent study by Fatt & Katz (1952) of spontaneous subthreshold activity at motor nerve endings. They found that at the tips of certain many-branching nerve endings, there are a large number of 'active spots' each giving rise to localized electrical pulses in the muscle fibre with which they make common contact. Here the statistical problem is to infer as much as possible about the individual 'active spots' from observations on the sequence of pulses in the muscle fibre.