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auto corr sum REVD

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Phil's dated note (7.31.13) for the August 2013 update of his Spectral Theory book, inserted in Appendix F on repeated sequences. It rewrites the autocorrelation r_s, the large-N limit of the average of y_n y_{n+s}, by splitting the sum into M segments of period P and using periodicity. The result is equation F.41b, a sum over one period. Equation symbols are partly garbled in the extraction.

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This is the Title PhL 7.31.13 This was installed in App F. rs ≡ limN→∞ [!Syntax Error, I yn yn+s ] = <yn yn+s> 1 . (32.16) (F.41a) For large N, we can replace the sum endpoints by -N = -MP and +N = MP ≈ (M+1)P where M is also large. The sum then has (2M+1)P terms, so rs ≡ limM→∞ [!Syntax Error, I yn yn+s ] . We now let n = IP + n' where n' takes values 0,1..P-1 and I = Int(n/P). Then the single sum above can be written, !Syntax Error, I yn yn+s = !Syntax Error, I !Syntax Error, I yIP+n' yIP+n'+s where this drawing shows how the sum is now a double sum where I denotes segments containing P points, and n' counts the points in a segment, We then process this sum using the periodicity (F.5) to get = !Syntax Error, I !Syntax Error, I yn' yn'+s = (!Syntax Error, I1 ) (!Syntax Error, I yn yn+s ) = (2M+1) (!Syntax Error, I yn yn+s ) . Inserting the expression into rs, we get this alternate form for rs rs = limM→∞ [{ (2M+1) (!Syntax Error, I yn yn+s ) } ] or rs = !Syntax Error, I yn yn+s = <yn yn+s>1 (F.41b)