MATHEMATICAL AX A LYSIS OF RANDOM NOISE 



71 



where the quadratic form is positive definite and | a | is its determinant. 

 A lu is the cof actor oi atu . Incidentally, these may be regarded as special 

 cases of 



[^ dx, ■■■ j clxjrZ OrsXrxA F (T, brxA 



7 r n-l-|l/2 ^+00 ^« 



2 /•/ 2 , 2-. 



fix + y ) 



(3.5-9) 



2^ Arsbrbs 



1/2^ 



i 



( 

 which is a generalization of a result given by Schlomilch.* 



3.6 DlSTRIBUTIOX OF MAXIMA OF NOISE CURRENT 



Here we shall use a result similar to those used in sections 3.3 and 3.4. Let 

 3'^be a random curve given by (3.3-1), 



y = F{ai '•' Gn ; x). 



(3.3-1) 



If suitable conditions are satisfied, the probability that y has a maximum in 

 the rectangle (xi , xi + dxi , ji , yi + dyi), dxi and dvi being of the same 

 order of magnitude, is " 



—dxi dyi I p(yi, 0, f)f d^ 



(3.6-1) 



and the expected number of maxima of y in a < x < b is obtained by in- 

 tegrating this expression over the range — =<= < yi < ^ , a- '^ xi < b. 

 /'(s> Vy D is the probability density function for the random variables 



^ = F(ai , • • • , c.v ; Xi) 



= ('!) 



\dx /i=xi 



(3.6-2) 



c 



^ ydx^),=,. 



*Hoheren Analysis, Braunschweig (1879), Vol. 2, p. 49-1, equ. (29). 



^- Am. Jour. Math.. Vol. 61 (1939) 409-416. A similar problem has been studied by 

 E. L. Dodd, The Length of the Cycles Which Result From the Graduation of Chance 

 Elements, Ann. Math. Stat., Vol. 10 (1939) 254-264. He gives a number of references 

 to the literature dealing with the fluctuations of time series. 



