1.8. Under the same hypothesis as III Theorem 7.3.3, prove that maxI

JT L 2k + 1 8x2 for x> O.

[HINT: There is no difficulty in proving that the limiting distribution is the same for any sequence satisfying the hypothesis. To find it for the symmetric Bemoullian case, show that for 0 < z < x we have

,/) {-z < min Sill S max Sill <

x -z; Sn = Y z}

= ;11 f {( n + 2kx n + y -z ) -(n + 2kx n -y -z ) } .
k=-oo 2 2 This can be done by starting the sample path at z and reflecting at both barriers o and x (Kelvin's method of images). Next, show that lim ,~{ -z.Ji1 < Sm < (x -z).Ji1 for 1 :s m :s n} n-+oo 1 00 {1(2k+I)X-Z j2kx-Z } ~ v2/2 = --D -e' dy ..ffii k=-oo 2kx-z (2k-l)x-z .
Finally, use the Fourier series for the function h of period 2x:
{-I hey) = ' +1, if -x -Z < Y < -z;
if -z < y < x -z;
to convert the above limit to 4 E 1 . (2k + 1 )".z [ (2k + 1)2 ".2]
This gives the asymptotic joint distribution of of whIch that of maxI