The Fluctuation Rate of the Chi Process

Cover The Fluctuation Rate of the Chi Process
The Fluctuation Rate of the Chi Process
Richard a Silverman
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-1 M, , -1 n+l, n+l ••• ''2n, 2n 2l^ ' O we find, returning to the variables g. And g. , that r n P(g-j^» • • • yg^jg-j^j • • . *gj^) (2. g2*«»«*gn) S(g^, g2, ... , g^) 3( ;^n'®l* • • • '®n-l» ^n*®l* * * ' '^n-1^ ^^^ n*®l» * * ' '®n-l^ ^^ "^n'^l* * * * »^n-l^ ^iz-^i%2> * * * »^r) e(Xn'®i, ... , Vi^ n - 7 - where the Jacobian J = l^n-l sin^^-^e sin"~%- ... Sin 9 - n n 1 . 2Pn-lf^r--f ®n. LP^^n'®l--^-l' Vl--^-l^ -00 -co which equals n (12) 00 * X c3&, exp J -00 \ i-Yr 1 dOgexp ^00 -2a'^R o.
exp 2
... • 2 -[ ^n "^ n — 5 5~n- 2ct'^ -2a'^R ^ o -* 2a2 . IZi sin ■^.
-2cV CO ... J d9^_^exp -00 "2*2 2 2« -^ 9, sin ©, . . . Sin^ « X n n-1 1 n-2 -2(r^R" Tl^ie integrals with respect to ®i>»*'»^n-l *^^ ^® ^°"® immediately. Their product is 2 " o % n n-1 "5" sin"" 9, sin'^"-^9^... Sin 9„ « 1 2 n-£ n-1 (. 2. O-2lfe")"^ n so that (12) becomes Pt^)K(^^)-("A) ^ 2 - (n/2) where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively, and k is the fvinction of t' defined by (16).


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