MATHEMATICAL ANALYSIS OF RANDOM NOISE 331 



The exponent in the expression (2.9-1) for the probability density may 

 be written out by using 



x'M-'x - E E ,^i XrX, (2.9-5) 



where Mrs is the cofaclor of jurs in M. 



Sometimes there are hnear relations between the .t's so that the random 

 vector r is restricted to a space of less than K dimensions. In this case the 

 appropriate form for the density function may be obtained by considering 

 a sequence of A'-dimensional distributions which approach the one being 

 investigated. 



If ri and rt are two normally distributed random vectors their sum ri -\- r^ 

 is also normally distributed. It follows that the sum of any number of 

 normally distributed random vectors is normally distributed. 



The characteristic function of the normal distribution is 



ave g"i^i+"22-2H \-izKxj^: 



r- ^ K K -1 



= exp -- E E Mr,2rzJ (2.9-6) 



L L r=l s=l J 



2.10 Central Limit Tiieorem 



The central limit theorem in probability states that the distribution of the 

 sum of A^ independent random vectors n + ''2 + • • • + ^jv approaches a 

 normal law as i\^ — * 00 when the distributions of n , ^2 , • • • r^ satisfy certain 

 general conditions. 



As an example we take the case in which ri , r2 , • • • are two-dimensional 

 vectors , the components of r„ being Xn and y„ . Without loss of generality 

 we assume that 



Xn = 0, y„ = 0. 



The components of the resultant vector are 



X = xi + X2-]r '" + Xn 



(2.10-n 



Y = yi + y2 •{■ • " -\- jN 



and, since ri , r^ , • • • are independent vectors, the second moments of the 

 resultant are 



)u:i = X^ = x\ + aI + • • • + x]i 



)U22 = F^= jf+ j!+ ••• +}J (2.10 2) 



/xi2 = X F = xiyi + .T23'2 4- • • • + .r.v3V 



^Incidentally, von Laue (see references in section 1.7) used this theorem in discussing 

 the normal distribution of the coefficients in a Fourier series used to represent black-body 

 radiation. He ascribed it to Markofif. 



^1 This case is discussed by J. V. Uspensky, "Introduction to Mathematical Probabil- 

 ity", McGraw-Hill (1937) Chap. XV. 



