2949775115

E[X(s,n,m, k)\X(r,n,m, k) = x] = a ^^(l//3)(_p)[l + (x/a)⁰]^p

p=o

<n(r

.1 'Ir+j-p/aJ

if and only if

F(x) — (l + {^} )    , x > 0, a, a > 0, /? > 1.

Proof Prom (4.1), we have

C_a-1

E[X(s,n,m,k)\X(r,n,m,k) — x] — -

(s — r — 1)! Ci— i (m + l)^s'

By setting u =    = (£§/§)£) ^from C¹-²) ^m (⁴-³)> ^{we obtain}

aCs-1

£J[X(s,n, m,/e)|X(r, n, m, fc) — x] = -

(s — r — l)!C_r-i (m + l)^s-r

X f\{ 1+ (x/<T)^,’}t.-^1/“ -

Jo

=    +!)--■ S^(1/«“¹¹ +

Jo

Again by setting t = u^m+1 in (4.4), we get E[X(s,n,m,k)\X(r,n,m,k) = x]

<r

t rn + l aC_a-i

= (a - r - iy.Cr-i{m + l)—'    ⁺

_r(_i^_+n__s)_r(s__T.₎

r(^ + »-r)

= (s-r- iyc‘Ii(m + 1) —^r    +

p—0

.. (m+l)*~T(ł-r) YĘZitir+j - (p/a))

and hence the relation in (4.2).

To prove sufficient part, we have from (4.1) and (4.2)

C_s-1

,m+l