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ON A FLUCTUATION THEOREM FOR PROCESSES
WITH INDEPENDENT INCREMENTS II
by
C. C. Heyde
Australian National University
and
University of North Carolina
Institute of Statistics Mimeo Series No. 586
July 1968
This research was supported by the Department of
the Navy, Office of Naval Research, Grant No. Nonr-855(09).
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C.
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On A Fluctuation Theorem For Processes With Independent Increments II
1.
By C. C. Heyde
Introduction and summary. Let ~(t), t~O,
~(O)
= 0 be a separable
stochastic process with stationary independent increments whose sample
functions are continuous on the right.
~-(t)
Write
~(s)
= sup
and
O~§~t
T = min[u:O~u~t; ~(u) = ~(t)].
t
The object of this paper is to establish
the following theorem:
Theorem.
The limiting distribution
lim
(1)
Pr(t
t-+<x>
_1
T < x)
t
= F(x)
exists if and only if
(2)
lim
Pr(t(u»O)du =
t-+oo
and then F(x) is related to a
(3)
E.I.
F(x) = Fa(x) =
si~~a J:V-(l~)(l-V)~dV,
Fo(x) = 0 if x<O, 1 if
O<a<l,
O~x~l,
.1
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~O
F1 (x) = 0 i f x<l, 1 i f x,al.
This theorem is the exact counterpart to a theorem of Spitzer ([4], Theorem 7.1)
7.1) for sums of independent and identically distributed random variables.
It contains as a special case the well-known arc-sine limit theorem for the
Brownian motion process.
An earlier version of the theorem was obtained by
Heyde [3] under the additional condition
J;t-1pr(~(t»0)dt<m, the
of which leads to Pr(~(t) = 0)
It should be remarked that T has
t
= 0,
t>O.
violation
the same distribution as Nt =~{u:O~u~t;~(u»O},~ denoting ordinary Lebesgue
measure.
This follows from the well-known corresponding result of Sparre-
Andersen for sums of independent and identically distributed random variables
by a straightforward limiting argument.
1. This research was supported by the Department of Navy, Office of Naval
Research, Grant No. Nonr-855 (09).
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Z
Z.
Proof of the theorem.
Xm, i' i = 1,Z,3, .••
of independent and identically distributed random variables defined for
integer valued
m>l
by
xm,i
and write
Sm,o = 0,
Sm,n =
= ~ (Z-mi ) - ~ (Z-m(i-l») ,
i~lXm,i
=
~(Z-mn),
n>l.
Write also,
T
= min[k : O~k<n, S
= mex S
]•
m, n
m, k 0:':'J <n m, j
Then, from well-known results of Sparre-Andersen,
Pr(Tm,n = k) = Pr(Tm, k
and for
'f
k=
= k)
Pr(Tm,n- k = 0) , O<k<n,
0.:s..t~l,
00
Pr( max S j
OS.j~k m,
= I: Pr(T
= O)t
k=o
m,k
'f
Pr{T k = k) t
k=o
m,
k
k
OOt k
= exp{I:
1k
.
Pr(S 1~0)}
m,1\.
= exp{'f!.kpr (S
>O)}
lk
m,k
,
(see for example Spitzer [4]), and we readily find upon taking generating
functions that for
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We introduce a sequence
O~v<l,
u>O,
n
(4)
00I: E (-uT
v (-nu)
e
m,n ) vn = (I-v) _1 exp {OO
- ~l n
l-e
Pr(S
n=O
n-
m,n
}
>0).
-Z-m z
-m
Now, put v = e
, u = Z z in (4), rewrite it in the form
-m
-m
-m
(1_e- 2 z)nI E(e- 2 STm,n)e- Z nz
o
= exp{-nI
and let
m+oo.
l
~ e-2-mzn(1_e-2-msn)Pr(~(Z-~»0)},
It follows from a straightforward argument in the spirit of
Baxter and Donsker [1] (see Lemma Z and the first part of the proof of
Theorem 1) that
(5)
oo -zt -sT
foo _1
-zt
E(e
t)dt = exp{- ot Pr(~(t»O)e
(l-e-st)dt}.
z oe
J
This result provides the basis for the proof of the theorem.
We now establish that the condition (Z) is sufficient for the existence
of the limit distribution (1) and that the form (3) follows.
In order to do
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this, we show firstly that under (2) and when O<A<l,
i1~
(6)
OO
Jat
_1
(
Pr
)
~(t»O
e
-zt (
l-e
-AZ~
] dt = a 10g(1+A).
Write
A(t) = t- 1 Jtopr(~(u»O)dU.
Then, integration by parts gives
f:t-lpr(E;(t»o)e-zt(l-e-Azt)dt = f:A(t)C(Z,t)dt,
where
C(z,t) = t
_1 -zt
e
[(l+tz) - (l+tz(l+A)e
-ZAt
l.
We note that C(z,t»O and l1mC(z,t) = O. Thus, for z>O,
oozO
JoA(t)C(Z,t)dt = f.e[A(t)-alc(z,t)dt + aJoC(Z,t)dt
00
-
00
= J:[A(t)-alc(z,t)dt + alog(l+A)
upon performing a simple integration.
Now, in view of (2) we can, given
€>O arbitrarily small, choose T so large that IA(t)-al<€ for t~T and then
IJ:[A(t)-alc(z,t)dt l
as z+Q sin a
~ J:IA(t)-alc(z,t)dt
+ €log(l+A)
l~
+
€log(l+A)
C(z,t) = O. The result (6) follows immediately.
z a
putting s = AZ in (5) where O<A<l and making use of (6),
(7)
11m zJooe-ztE(e-AZTt)dt
z a
a
=
(l+A)-a.
Now,
where
A..
-lK
f_z)k+l foo
= -~
kl
(z)
k
e-ztE(T )dt,
a
t
so that from (7),
00
k
lim
kL A A.. (z)
z+O =0 -lK
and consequently,
=
(l+A)-a
=
E Ak(-a)
k'
k=O
Then,
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4
(8)
But, ET~ is monotone in t so, using Theorem 4, 423 of Feller [2], it follows
from (8) that as t+oo,
E(t-1Tt)k ~ (_l)k(~a) , k>O.
Furthermore it is easy to verify that
(-l)k(~a)=J:xkdFa(X)'
where F (x) is given by (3) and, since the moment problem in this case has a
a
unique solution, the proof of the sufficiency part of the theorem is complete.
Finally, suppose that t
t
-1
ETt~
-1
for some O<a<l as t+oo.
T has a proper limiting distribution.
t
Also, upon differentiating with respect to
s in (5) and putting s=O,
J
~ -zt
z oe
l~ -zt
E(Tt)dt =
Consequently, noting that ET
t
e
Pr(~(t»O)dt.
is monotone in t, we have from Theorem 4,
423 of [2] that
zJ~e-ztpr(~(t»O)dt~
as
z~
Then,
o
0, and the condition (2) follows from Theorem 2,421 of [2].
completes the proof of the theorem.
This
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REFERENCES
[1]
Baxter, G. and Donsker, M. D. (1957).
On the distribution of the
supremum functional for processes with stationary independent
increments.
[2]
Trans. Amer. Math. Soc. 85 73-87.
Feller, W. (1966).
An Introduction to Probability Theory and its
Applications Vol. II.
[3]
Heyde, C. C. (1968).
Wiley, New York.
On a fluctuation theorem for a class of processes
with independent increments.
Technical Report No.9, Statistical
Laboratory, Univ. Manchester.
[4]
Spitzer, F. (1956).
A combinatorial lemma and its application to
probability theory.
Trans. Amer. Math. Soc. 82 323-339.
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