COMMUNICA TION IN PRESENCE OF NOISE 77 



When the expression (5-17) for (2 — 22) is put in (5-18) an expression for 

 /"(:;) is obtained. This expression, together with 



log T{q + 1) = (g + 1/2) \ogq-q+ (1/2) log Itt + 0(l/g), 



enables us to write the argument of the exponential function in (5-15) as 

 q log 2r - (1/2) log lirq - Q(a, /3) + 0(q-'i'- log^/'-^ q) where Q{a, 0) denotes 

 the quadratic function 



(5-19) 



Qia, 13) = [(1 + rYia- -f 13'-) - 2r(l + r)al3]D 

 D = \/[2q{\ + 2r)] 

 Similar considerations show that 



(^2 _|. .)-i/4 = ^-1/2(1 + 2r)-i/2[l + 0(^-1/2 logi/2 q)] (5-20) 



When the above results are gathered together it is found that (5-12) 

 may be written as 



p,{u, xo) dii dxo = D, exp [-Q(a, (3) + 0(q-''-' log^/^ q)] da d(3 (5-21) 

 where 



Expression (5-21) is valid as long as | a | and | iS | do not exceed 



(2q log qr\ 



Expression (5-21) differs from the one predicted by the central limit 

 theorem (and (5-2) and (5-3)) in that it is not quite centered on the average 

 values Xo, «, which correspond to a = 1, /3 = 1, respectively. Also, q enters 

 in place of ^ -f- 1. However, these differences amount to 0{q^^- log^'- q) 

 at most, as may be seen by putting a — 1 and (3 — 1 for a and /3 in (5-19). 



By using relations (5-6) and (5-21), it may be shown that 



Prob. (PiQ, ••• ,PkQ> PoQ) 



r« r* (5-23) 



= / da d0D,e~'''-''['L-Pixo,u)]'' + O{q-'"log'''q) 



J—q J—q 



where it is understood that .vo and u in P(xo, u) depend on a and /3 through 

 (5-11). The term 0{q~^'- log^'- q) in (5-23) represents the sum of three con- 

 tributions. The first is Ri in (5-6) which is 0(1 g). The second arises from 

 the fact that when the factor exp [^{q~^'- log^'- q)] in (5-21) is neglected in 

 integrating (5-21) over -^ < a < I, - C < fi < /", where /" = 

 {2q log qY'-, the resulting integral is in error by 0(^~^'- log^'- q). The third 

 is due to the contributions of the integral from the region \a\> (■,\^\> (. 



