I've been looking at detector coincidences and tried to find what general limits apply to coincidences. I was surprised how simply the calculation works out. My question is whether it is correct and where can I find similar stuff ?(adsbygoogle = window.adsbygoogle || []).push({});

Consider two binary sequences produced by random processes where the probabilities of getting 1 are ##p_1## and ##p_2## respectively. Now assume that the number of 1's in the streams is ##1_n \rightarrow Np_n## as ##N \rightarrow \infty##.

If we know the counts ##1_n,\ 0_n=N-1_n## in our sequences then by a permutation argument it is clear that the maximum number of (0,0) coincidences can not be greater than the minimum of ##0_1=N(1-p_1)## and ##0_2=N(1-p_2)##. Similarly the maximum possible (1,1) coincidences is the least of ##Np_1## and ##Np_2##. Assumimg ##p_1<p_2## this gives a total for the (0,0) and (1,1) coincidences of ##S_{12}=N(1-p_2+p_1)##. There are no permutations which give a greater total than this.

The maximum possible correlation between the streams is given by ##\mathcal{C}_{12}=(2S_{12}-N)/N## which gives ##1-2(p_2-p_1)##.

From this one can write for the maximum possible correlations between 4 streams ( assuming ##p_1\leq p_2 \leq p_3 \leq p_4##).

##|\mathcal{C}_{12}+\mathcal{C}_{23}+\mathcal{C}_{34}-\mathcal{C}_{41}| \leq 2##

the ##p_n## terms conveniently cancelling.

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# Correlation limits for binary variates

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