MATHEMATICAL ANALYSIS OF RANDOM NOISE 309 



is arbitrary but fixed. The expression analogous to (1.4-5) is the 2N fold 

 integral 



dui "I dvy (1.7-11) 



00 J— 00 



exp [ — i(oiWi + • • • + bffi'x) — vT -{- vTE] 

 where 



E = —- I dd exp i X («n Cn + Vn Sn) COS ud + {vn Cn " w„ 5„) sin nO 

 zir Jo L "=i J 



(1.7-12) 



in which Cn — iSn is defined as the Fourier transform (1.7-4) of F{t). 



The next step is to show that (1.7-11) approaches a normal law in 2N di- 

 mensions as J/ — > 00 . This appears to be quite involved. It will be noted 

 that the integrand in the integral defining E is composed of N factors of the 

 form 



exp [ipn cos {7td — }pn)] 



= /o(pn) + 2i COS {nd — \pn)Jl{pn) " 2 cos {2nd — 2)/'„)J2(pn) + • • • 



where 



2 / 2 , 2 \ /^2 io2\ 2 2/2 1 2\ 



Pn = (Wn + Vn){Cn + 0„) = ^a„(Mn + Vn). 



As J/ becomes large, it turns out that the integral (1.7-11) for the prob- 

 abiUty density obtains most of its contributions from small values of u and v. 

 By substituting the product of the Bessel function series in the integral for 

 E and integrating we find 



£ = n JoiPn) -\-A+B^C 



n=l 



where A is the sum of products such as 



-2i cos {\pk-^t — \pk - ^()J\{pk)J\{pt)J\{pk+() times A^ - 3 /o's 



in which Q < k <l and 2 < k -\- 1 < N. Similarly B is the sum of products 

 of the form 



— 2i cos (i/'2fc - 2\l/k)Ji{p2k)Ji{pk) times N — 2 Jo's 



C consists of terms which give fourth and higher powers in u and v. There 

 are roughly N^/4 terms of form A and N/2 terms of form B. 



Expanding the Bessel functions, neglecting all powers above the third and 



