~
· ..
J
SOME RESULTS USING GENERAL MOMENT FUNCTIONS
by
Walter L. Smith
University of North Carolina
Institute of Statistics Mimeo Series No.
554
--
This research was supported by the Department of the
Navy, Office of Naval Research. Grant NONR-855(09).
DEPARTMENT OF STATISTICS
University of North Carolina
•
Chapel Hill, N. C•
1.
Introduction
The work of the present note originated in a desire to prove
Theorem 1 below, which, in a somewhat more restricted version, was
annololIlced by the author during the Chapel Hill Symposium on Congestion
Theory (Smith and Wilkinson (1965), pp. 132-133).
The proof then available
was needlessly complicated; it will now follow from a useful general
theorem on generating functions (Corollary 3.1, below).
In Smith (1967) a class ~ of "moment functions" M(x) is introduced
as a means of dealing with problems involving a random variable X, say,
when it is given that the highest absolute moment of X known to be finite
is not necessarily an integral power of X.
given that tlxl3/2 10g+lxl <
00.
For example, one might be
In Smith (1967) the use of such informa-
tion in the expansion of characteristic functions and in certain renewaltheoretic problems is discussed.
A rather more fundamental property of
a large class of these functions M is as follows.
If Xl' X2,···, are
independent and identically distributed random variables such that
o
<
jl
=
& Xn
<
00
and i f S = Xl +. H+ X ' then provided eM( Ixl) <
n
n
follows that £M( IS
n
I) -
M(nll) as n
-+
00.
00
it
This result is described in
detail in Theorem 2.
Theorem 3, below, is then an easy consequence of Theorem 2.
It
concerns the finiteness of gM( IsNI) when N is a non-negative integervalued random variable independent of the {X}.
n
When the {X } are themn
selves also non-negative integer-valued random variables we are led at
once to Corollary 3.1, below, concerning a useful closure property of
2
probability generating functions of probability generating functions. - As
explained, it is from this Corollary that we deduce Theorem 1, the original
object of the present work.
We now state:
Theorem 1
(i) Let {X } be an infinite sequence of independent and identin
cally distributed random variables. with common distribution function
F(x) = p{X < x}, such that 0 < eX < +
n n
00.
(ii) Write Sn = X1 +•..+ Xn' as usual , and Tn = max{S l' S2' ••• , Sn }.
(iii) Let v. > 0 be an integer and M(x) be a non-decreasing function
of x > 0 such that M(x)/x is non-increasing for all large x.
Suppose
M(O) > 0 (and note that M(x) may be a constant identically).
Then any
one of the following three statements implies the other two:
<
(1.1)
,
00'
-00
E
(1.2)
n=l
n(V-l)M(n)p{S
< O} < 00'
n'
(1. 3)
In order to prove this result on random walks, which, incidentally
is of some relevance to the theory of queues, we prove the following
Theorem 2.
Theorem 2
This theorem would appear to be of general interest.
Suppose the conditions and notations (i), (ii) and (iii) of
Theorem 1 hold, with the additional proviso that
].l
= ex
n
<
00.
3
If, however, 11 =
IXn
I
= 00 (which can be the case only when v = 0)
one can infer from eM( XII) <
00
I I)
only the weaker conclusion IM( Sn
= O(n).
From Theorem 2, as already explained, we can easily deduce the
following theorem concerning momen ts of sums of random numbers of random
variables.
Theorem 3
(i) Let {X } be an infinite sequence of independent and identin
cally distributed random variables such that 0 <
B (x) = p{S
n
Ixn
<
00
and write
< x} for n = 1,2,··.; B (x) = p{O < xl.
n -
0
(ii) Let {c } be a discrete probability distribution (!..~. c > 0,
n
n 00
all n, andE
0
c
n
= 1) such that c
0
< 1.
(iii) Let the d.f. A(x) be thus defined:
A(x)
00
= n=o
E
c B (x)
n n
(iv) Let v and M(x) be as in Theorem 1 (iii).
Then a necessary and sufficient condition to ensure that
+00
(1.4)
J
IxlvM(/xl)dA(x) <
-00
is that both the following statements be true:
00
4
+00
J
(1.5)
IxIVM(lxl)dB1(x) <
00;
_00
~
(1.6)
n~l
= O.
If. for the case v
except for the fact that
nVM(n)c
n
<
00.
it should happen that (i).···.(iv) hold
Ix n = +00
then one can still infer that (1.4) is
00
true if (1.5) is given, together with the condition Ll nc <
n
If v and M(x) are as in Theorem 1 and if n(s)
00.
= L~nSn,
0
<
s~ 1,
is a probability generating function, then we shall say that n is in the
c1ass1 (v, M) if
An immediate corollary to Theorem 3 is the following:
Corollary 3.1
Let v and M(x) be as in Theorem 1 (iii) and let a(s),
S(s), yes) be probability generating functions, 0
S(s)
n
= Lobn
s , o
that Ln
nb
00
00
<
00,
and that a(s)
~
s
~
= yeSes»~.
1.
Suppose that
Then
a~(v,
~T
M) if
and only if both S and y belong to, (v, M).
If, for the case v
00
= 0, it should happen that L1nbn =
00
one can
still infer that ae'-(O,
M) if it is given that Se~(O,
M) and Loonc
<
~
T
1 n
00.
We close this introduction by mentioning that in the course of
proving Theorem 2 we establish the following:
Corollary 2.1
as n
~
00.
Under the conditions of Theorem 1. for any e > 0,
5
2.
On
moment functions and some preliminary lemmas.
class~
In Smith (1967) we have introduced a
A function M(x) belongs to»J
~
(ii) M(x)
1, all x
~
(i) it is non,.,.decreasing in [0,
if:
0; (iii) M(x + y)
If M£~and, additionally, (iv) M(2x)
*
M£~.
if
We say a d.f. F(x) is in the
t.: Ix I
I I) dF(x)<w.From
v M( x
of moment-functions.
~
M(x)M(y) for all x, y
= O(M(x»
00);
~
O.
for all x > 0 then we say
class~(v,
M), for any real v
~
0,
the point of view of defining a class P (v, M),
the moment function M(x) can be replaced by any non-decreasing function
N(x) such that N(x»( M(x) as x
-+00
there exist positive constants 0 and
•
all large x).
X
~
-+ 00"
~
such that o<j>(x)
1jJ(x)
means that
~ ~1jJ(x)
for
Thus we are led to introduce an extended class EX»* in such
*
N(x)£E~
a way that a function
N(x)
("<j>(x»)( 1jJ(x) as x
M(x) as x
-+ 00.
if we can find a function M(x)£.:ttl* such that
We can then say M(x) represents N(x) in1,;t1*.
Suppose that for some
~ >
0 and some A
>
0 the function N(x)
satisfies the following:
(i)
(li)
(iii)
(iv)
N(x) is non-decreasing for x >
N(~)
> 0;
N(x + y) : AN(x)N(y) for all x
N(2x)
~;
m
O(N(x»
for all x
> ~,
y
> ~;
> ~.
Then we may suppose, with no loss of generality, that
M(x) = AN(x) ,
= M(~) ,
x >
~,
x <
~;
AN(~) >
1 and define
6
*
to~.
the function M(x) so defined belongs
Thus a function N(x) satisfying
(i), (it), (iii) and (iv) above belongs to EJ71* •
~
Now suppose that, for some 6
is non-increasing for all x
and define N\I (x)
= xVM(x)
2, M(x) is non-decreasing and M(x)/x
6, and M(6) > O.
~
for all x > 6.
above are satisfied by N\I(X).
Let v > 0 be a constant,
It is clear that (i) and (ii)
Further, if x
+
• (x
~
6 and y
y)
M(x)M(y)
~
6,
{M(X + Y) }
x + Y
But, since M(x)/x is non-increasing,
(x +
y
)
f M(x
+ Y»)
(x + y)
<
-
x {M(X)/
x
•
so that
N)x + y)
(1
l)V { 1
1
N (x)N (y) .::: i' + y
M(x) + M(y)
v
\I
J
2
.::: M(6) ,
since x > 6, Y > 6.
Thus N(x) satisfies (iii) above.
N (2x)
v
N (x)
v
<
=
\1 M(2x)
2 M(x)
2(\1+1) ~ {M(2X) }
M(x)
2x
< 2(v+l)
_x_·f
~J
M(x)
x
Finally, for x > 6,
7
Thus N (x) also satisfies (iv) and so belongs to
\I
function M (x) e:1»* which represents N (x).
\I
\I
* and
~
there will be a
We shall make frequent use of
these functions M (x).
\I
With these preliminaries set aside we can now prove the following.
Let k > 1 be an integer and let M(x) e: 1.lJ.
Lemma 2.1
Suppose Xl' X2 , · · · , X
k
are independent and identically distributed random variables and write
Sj
= XI +.•.+
IM( IXII) <
Proof.
Xj for j
= 1"
2 •••
k
,.
Then ~M( ISk I>
<
00
if and only i f
00.
Clearly M(ISkl)
~ M(~~IXjl), and so, by a property of functions in~
k
M(ISkl) < j,DI M<!Xjl)·
Therefore
and part of the lemma is proved.
If P{X I > O}
trivial.
= 1,
or if P{X 1
~
O}
Suppose therefore that both P{X 1
=1
~
the rest of the proof is
O} > 0 and P{XI < O} >
Then both P{Sk_1 ~ O} > 0 and P{Sk_l < O} > O.
o.
The finiteness of'M(lskl)
then implies that, for example,
is also finite.
However, if both Sk-l and ~ are negative then IXkl
<
ISkl
and so
must be finite.
But Xk and Sk-l are independent.
Thus &{M(IXkl)l~ < O}
8
is finite.
Similarly one can deal with &{M(IXkl)IXk ~ OJ, and the lemma
follows.
Lemma 2.2
If. for large x. L(x) is non-increasing and xL(x) is non-
decreasing. then
lim
£-1-0+
lim
£-1-0+
Proof
If we set M(x)
=
L(l + £
L(n)
lim sup
n)
= 1
n-+oo
lim inf L(lL~n) n)
=1
n-+oo
xL(x) , then
M(l + £ n)
M(n)
> 1,
which implies that for all n
1 > L(l
+
£
L(n)
n) >
1
- 1 +
£
The lemma follows at once from these inequalities.
9
3.
Proof of Theorem 2
Let us write M (x), v > 0, for any member
v
M (x)
v
x
x vM(x)
find a Kl
>
as x
0, K2
-+
>
of~
it
such that
Suppose that the integer v > 1.
00.
0, such that, for all large x,
(3.1)
For large R > 0 and small e > 0, define events
(e) -
{IS n In
B (R) _
{I Sn In
A
n
n
~I <
~
-
I
e}
> R}.
Further, for n = 1,2,···, let
M
b. (R, n)
v
=
J
B (R)
v
(IS n I)
M (n~)
dP.
v
n
Then
b. (R, n)
v
<.-::... I
-
K2n~
ISn I
B (R)
n
dP,
Then we can
10
by a defining property
*
of~.
Hence we have
!J. (R, n) < Kl
-v
- K~ ].J
I
M
(IS
n-l
M (n)
V-I
Mv ( IXn I)
V-I
B (R)
I)
dP
].J
n
1
>"2 nR
>
t
nR
(3.2)
Because X and S
n
n-l
are independent we have
(3.3)
1
M (IX l)dP,
v
1
> 2" nR
using the fact that Xl and X are identically distributed.
n
In a similar way we find
(3.4)
For integer A > 0, let"A be the hypothesis that the following two
statements are true:
11
~A(R,
(ii) For all sufficiently large R,
n)
-+
The inequalities (3.2), (3.3) and (3.4) show that if
0,
as n
-+~.
eM) Ixll) is finite
and if~ (v-I) is true, then
(3.5)
~
for all large R.
v
(R, n)
0,
as n
-+ ~,
Furthermore,
1Mv (IS n I)
(3.6)
-+
M (R
v
+
]..I n)
M (n]..l)
v
But, for some K4 > 0,
M (R
v
+
]..I
n)
<
K4(R +]..1)
v
M(R + ]..I n)
v
M (]..In)
v
]..I M(]..In)
(3.7)
since M(x)/x is non-increasing.
From (3.5), (3.6) and (3.7) we see that
tM (IS I) = O(M (n]..l»,
v
n
v
Thus ill A is true ifdt(A_I) is true and
show
H
v
that~
o
as n -+~.
Mv(lxll) <~.
Plainly we desire to
is true, from which it would follow (if 1M) IXII)
<
~) that
is true.
Let us write M(x)
of x.
Then
= xL (x) ,
where L(x) is a non-increasing function
12
Is n IL( Is n I)
dP
nj.JL(nj.J)
J
t. o (R, n) =
B (R)
n
I
L(R - u n)
- j.JL(nW
(3.8)
<
I Snnl
dP.
B (R)
n
= 1 "2
Let us define, in a familiar way, for n
X
_> 0, X+ = 0
j.J-
=
n
n
if X
n
We then have Xn
< O.
IX-,
both necessarily finite because
n
and so on.
We then have (R »
B (R)
n
:/
dP
- X ; j.J+
n
IIXn I
I
=
J
(R)
s- > s+
n
n
B
-<
J
:
dP
/:1
I
-
+
s+ > n(R+j.J)
n -
>
0
...!l dp > (j.J+ -
d
£
1, so that
S+
lim inf
n-+oo
n
<
£
But e(s+/n) = j.J+, and hence, for (R + j.J) > j.J + +
n
lim sup
n-+oo
= Xn
= Ix+n
and
J
dP
s- > n(R-j.J)
n
By the weak law of large numbers, for any
-+
n
n
s+
p{ls:/n - j.J+, ~ £}
X+
< <X>;
,:n/ dP +
B (R)
n
s+ > sn - n
(3.9)
,
j.J),
S
II
= X+n
••••
S+ > n(R + j.J)
n -
£,
-
S
..1l dP
n
if
13
The arbitrariness of e tells us that
n -+-
00.
A similar argument will deal with the second integral in (3.9) and show
that
Snn\ dP
J I
B (R)
-+-
0,
n -+-
00.
n
This result coupled with the observation that L(R -
~
n) <
L(~n),
for R »
enables us to deduce from (3.8) that
A (R, n)
o
for R sufficiently large.
-+-
n -+-
0,
The proof that' M( Is
00,
n
D=
0 (M(n~»
now follows
easily from our inequality (3.6) with zero substituted for v.
Thus hypothesis~ 0 holds and consequently so does~v'
We shall now
use the weak law of large numbers to infer that, for any e > 0,
P{A (e)} -+ 1 as n -+
n
f
00.
We then have
B (R)n - A (e)
n
n
M (IS
v
n I)
Mv (n~)
M (R
<
~ n)
v
<
-
by an argument we have already used.
+
-v':"M--:(-n~~)-- P{- An (d }
K4
+
(~
~
(v+1)
P{-A (e)}
n
Therefore, using the fact that
~,
14
~
\)
(R, n)
~
0 as n
~ ~,
we can infer that
&Is n I~(IS n I) =
( 3.10)
f
Isn I\)M(lsn l)dP + oeM\) (n~».
A (E)
n
We first deduce from (3.10) that
(3.11)
<
lim sup
n~
(u
+ £)(\)+1) L(u +
E n)
~(\)+1) L(~n)
< 1,
in view of the arbitrariness of
E
and Lemma 2.2.
We secondly deduce from
(3.10) that
, Is I\)M( Is I)
(3.12)
lim inf __n:::-_ _.::n~
n~
(n~)\) M(n~)
> lim inf
n4=
(u - £) (\)+I)l(u -
-
-~(\)+1) L(~n)
E
n)
- P{A (E)}
n
> 1,
by the arbitrariness of
E,
the weak law of large numbers, and Lemma 1.
The
theorem follows from (3.11) and (3.12).
It will be clear from our proof that Corollary 2.1 is also established.
15
4.
Proof of Theorem 3.
Let k be the least integer such that c
k
>
O.
Then it follows
from the assumption A~(v, M) that BkEP(V, M); in other words, we have
From Lemma 2.1 we then have that
If we may suppose 0 < eX l <
CIO,
we can now appeal to Theorem 2 and deduce
that
From this asymptotic relation and the assumption A~v, M) we are easily
led to the conclusion
The necessity part of the theorem then follows from the easily proved fact
that M(nll) >< M(n).
It should also be obvious at this stage how to prove
the sufficiency part of the theorem.
Finally we must deal with the case v
= o and
the monotonicity of M(x) ,
+.•.+
Ixn I)
jtX
n
=
CIO.
We have, by
16
But, sinceM(x)/x may be supposed non-increasing for all x
~
0, we have
inequalities like
and thus discover that
Mcls n I) -< M(IX11) +- - -+ M(/Xn I)Hence, if !M(IX11) < co, it follows that &M(lsnl)
is complete_
=
O(n) and the theorem
17
5.
Some results on random walks; proof of Theorem 1.
We continue the assumption that {X } is a sequence of independent
n
and identically distributed random variables; we write F(x) = P{X
for their common d.f. and S = Xl +..•+ X , as
n
n
We suppose 0
<'
usual~
l
~
x}
for the partial sums.
Xl .::: +>0 •
The following Theorem A is an easy consequence of Theorem 6 of
Smith (1967).
In that paper it is supposed that tX l is finite whereas we
are here making no such assumption.
If tX l = +>0,
truncation argument suggests itself readily.
hOTi~ever,
It will be interesting to see
how much is a consequence of this Theorem A.
Theorem A Let ep(x)
I:: ~
'*
and suppose
If, for any cons tant y < &X l ' we write
A(x)
then A(O) <
If 'Xl =
~
~
'f 1.
P{S
= n=l
n
n
- n\l _< x},
0
and
we may take y arbitrarily large.
a suitable
18
We note that A(x) is obviously a non-decreasing function of x
and that ~(Ixl) A(x)
-+
0 as x
-+
-~.
~
0
Thus, after an integration by parts,
the conclusion of Theorem A can be rewritten:
This means
<
~
so that
and 1!. fortiori
~(ny)
-
~(O)
'f - - - n- - -
(5.1)
n=l
v
Let M (x) represent x M(x) in
v
enunciation of Theorem 1.
P{S
< O} <
n -
•
m,
where
~.
M(x) is as defined in the
Then we deduce from Theorem A and (5.1) the
following:
Result 1
(1.1) implies (1.2).
Elsewhere in Smith (1967) it is shown that the assumptiontxl > 0
implies the existence of constants p > 0, n > 0, such that
P{Sn
~
O} > pnP{X l
<
-nn}.
Thus, if (1.2) holds, we must have
'f
n=l
M ( 2nn)P{X
v
1
<
-nn}
<
~,
19
since M (2nn)
v
)C
M (n)
'f
n=l
as n
v
{ r=l
~
Therefore
-+~.
Mv(2rn)] P{-(n + l)n < Xl < -nn} <
~.
But, for large n,
n
~ Mv (2rn)
r=l
>
-
l
2 (n - l)nMv (nn).
Hence
We have therefore established:
Result 2
(1.2) implies (1.1).
Let us now define, for n
1T.
n
=
= 1,2,···,
P{T <
n -
O}
Then a familiar identity of Spitzer (1956) states that, for 0 < s < 1,
(5.2)
From this identity, on comparing coefficients, it is clear that
for all n.
Result 3
1T
Thus we have:
(1.3) implies (1.2)
Finally, suppose again that (1.2) holds.
<
~
Then we can set
n
>
A
n
20
and
S(s)
= n=L 1 ().n A-1 ) sn, o
00
< s < 1,
will be a probability generating function in the classf(v, M).
Further-
more
yes) = e
A(s-1)
is the familiar Poisson probability generating function and yes) is plainly
a member of~(v, M).
From Corollary 3.1 we can infer that yeSes»~ is
another member of 1'(v, M) if only we can be sure that the "mean" of S(s)
is finite,
~.~.
that
00
L
n=1
n).
n
<
00.
This is the same as the statement
00
L
(5.3)
When v
n=1
P{S
<
n -
o} <
00.
1, (5.3) is a consequence of (1.2), which we are assuming true.
~
Thus, when v
Result 4
~
1, we have
(1.2) implies (1.3)
However, we must also prove Result 4 when v = O.
(5.3) is, in general, false.
00
function yes) = L C s
o n
n
In this case
However, if we expand the Poisson generating
00
it is obviously true that L1 nC <
n
case v = 0 of Corollary 3.1 completes the proof of Result 4.
Theorem 1 follows from Results 1, 2, 3, and 4.
00.
Thus the
References
W. L. Smith (1967), A theorem on functions of characteristic functions and
its application to some renewal theoretic random walk problems,
Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics
and Probability, Vol. II, Part 2, pp. 265-309, Berkeley, University
of California Press.
W. L. Smith and W. E. Wilkinson (1965), Proceedings of the Symposium on
Congestion Theory, Chapel Hill, University of North Carolina Press.
F. Spitzer (1956), A combinatorial lemma and its application to probability
theory, Trans. Amer. Math. Soc., 82, 323-339.
UlCWlSD'Dm
,
Sertlrit'·
Classification
~
DOCUMENT CONTROL OAT A • R&D
fSf'clJrit\· cl/L<"-:;ification of fit/C!, "()d~' (.f abstract and indl"?xing annotation must be entered when tile overall report Is classified)
ORIGINATING
1
ACTIVITY
za,
(Corporate arJthor)
REPORT SECURITY CLASSIFIC4TION
---
zb. GROUP
,
REPORT
Tt.._ E
TI
SOME RESULTS USDa GIBltAL MOMEII'! J'UETIOIS
4. DESCRIPTIVE NOTES (Type o(report and incfusive dates)
Technical Beport
AU THOR\S)
~j
'First name. middle initial. last name)
Smith, Walter taws
REPORT DATE
"
7a. TOTAL NO. OF PAGES
17b. NO. OF REFS
BoveJlber 1967
Ba.
CONTRACT oR GRAt..JT NO.
h. PROJEC T NO.
9a. ORIGINATOR'S REPORT NUMBER(S)
.00-855-(09)
Institute of statistics JIi_o Series
-.alter 554
RR-003-05-Ol
c.
9b. OTHER REPORT NOIS) (Any other numbers that may be assil/ned
thIS report)
,f.
t-:--- STRI:-'lUTION
STATEMENT
Distriltlltion of the docu.at is unlimited
11
_~IJPPLEMFNTA'~'Y
,,<OTES
lZ. SPONSORING MILITARY ACTIVITY
I..ogistics aa• .athematical Statistics
Braach ottice of Baval Research
WuhiDItoll, B. c. 20360
1'
A P 5 it, ACT
Let (~) De a sel(Ueace of independent &I1d identical17 distributed random
variables sllch that 0 < '" •
eXa<
+
00
and write Sn· ~ +12 +••• + xn '
be an integer and let M(X) De a IlOD-decreu1Dg tunctioll of x
M(x)/x is :D.OD-increasing and M(O)
>
o.
follows that elsnIYM(lsal) - (at£) YK(IlJi)
~
-.-.
-
y>O
0 such that
Then if el~IYM(IX11) <
UI1
Let
00
ad Il<ao:it
It ~- (Y • 0) then
I> :: 0 (n). A. variet7 of results stem from this main theorem (neorem 2),
D
concerning a closure propert7 of probabilit7 geaeratiag f'UDctions and a rudom
eM(ls
walk result (neorem 1) cooected with queues .
•
I
Sectlrit,· Classification
.' .. ..
Security Cla..mcaUon
14.
LINK II
LINK.
Kay WO"D'
"OLa
WT
"OLa
WT
·-OJ
LINK
RO.LIE
c:
WT
:Renewal !heo17
I
I
.•
Jlaadom Walks
I
IbMJltl
ProDabilit7 Generating I'lmctioal
.
1
Security Classification