‘A note on the expectation of products of autocorrelations’

Abstract

The seventh formula m E ( r i 2 r k 2 ) = ( n − i ) ( n − k ) + 12 ( n − k ) − 8 i requires i + k ≤ n ; therefore the words “and seventh’ should be added to the assumption in lines 4–5 of p. 529. However, with the notation < x > = { x ( x ≥ 0 ) , 0 ( x < 0 ) , the whole assumption can be removed. For any i<j<k<l< we obtain m E ( r i r j 2 r l ) = 6 〈 n − 2 j 〉 + 2 i + 6 〈 n − i − 2 j 〉 ( i + l = j + k ) , m E ( r i r j 2 r l ) = 6 〈 n − 2 j 〉 + 2 i + 6 〈 n − i − 2 j 〉 ( i + l = 2 j , l = i + 2 j ) , m E ( r i 2 r k 2 ) = ( n − i ) ( n − k ) + 4 ( n − k ) + 8 〈 n − i − k 〉 . The rest of our formulae remain. We are grateful to 0. D. Anderson for bringing these points to our attention; he suggested (Anderson, 1982, p. 11) the use of < >.