ON A WALD'S EQUATION AND AVERAGE SAMPLE NUMBER IN A SEQUENTIAL SIGNAL DETECTION By

It is shown that in detecting sequentially a deterministic signal 0(0 in white noise 72(0 a similar identity (iii) in theorem 2.1, to the Wald's holds concerning a stopping time r determined by making use of a likelihood ratio. It is also shown that r has finite moments of any order under quite weak conditions over the signal. The exact A. S. N. E{y} in a constant signal case has been obtained and given by (2, 8). It is also considered a detection problem of a constant signal OW a in a colour-ed noise based on a sub-optimal statistic which become optimal when the noise were white. Similar properties of a stopping time r to those in the white noise case have been obtained in theorem 3.1. § 2. Detection of a deterministic signal in a white noise. We consider the following detection problem of a signal OW in the white noise 72(0; H; x(t) = W(t) H; x (t) = m(0+W(t),(2.1) where m(t) = 0(s)ds is the integrated signal and { W(t), 0 t < oo} is the Wiener 0 process which is considered to be the integrated form of the white noise 17(t). By Ho we mean that there is no signal in the (integrated) observationx(t) whose distribution is induced from the Wiener measure P0 and by Hl the observation x(t) is the sum of the signal m(t) and the noise W(t) whose distribution is induced by P1, e. a shift of Po by m(•). In order for the detection problem (2.1) to be non-singular, we assume that 0(•) is square integrable on each finite interval [0, t], 0 t < oo. Let us pu