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SOME RESULTS USING GENERAL MOMENT FUNCTIONS
by
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Walter L. Smith
University of North Carolina
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Institute of Statistics Mimeo Series No. 554
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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.
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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
announced 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.
2
given that' Ix1 3/ 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
< II =8x
n
<
00
and if S = Xl· + ••• + X , then providedeM(IXI) <
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 eM( 1sNI) when N is a non-negative integervalued random variable independent of the {X}.
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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 t it is from this Corollary that we deduce Theorem It the original
object of the present work.
We now state:
Theorem 1
(i)_ Let
{X }
_
_ _ _n,
be an infinite sequence of independent and identi-_
cally distributed random variables. with common distribution function
F(x) = p{X < x}t such that 0
n (ii) Write S
n
= X1
<
eXn
<
+
00.
+...+ Xn t as usual t and Tn
= max{S l'
S
2'
••• S }.
t n
(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
MeO) > 0 (and note that M(x) may be a constant identically).
Then any
one of the following three statements implies the other two:
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(1.1)
_00
(1.2)
00
n~1
(1.3)
\I
n M(n)P{Tn
<
O}
<
00.
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 I hold. with the additional proviso that II
= exn
<
00.
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If, however,
l.I
= IXn = (which can be the case only when
<Xl
one can infer from tM(lx11) <
<Xl
v = 0)
only the weaker conclusionIM(ISnl) = O(n).
From Theorem 2, as already explained, we can easily deduce the
following theorem concerning moments of sums of random numbers of random
variables.
Theorem 3
(i) Let {X 1 be an infinite sequence of independent and identin
cally distributed random variables such that 0 < Ix
n
B (x)
n
=
p{S
xl for n = 1,2,··.; B (x) = p{O
<
n -
0
<
<
<Xl
and write
xl.
(ii) Let {c 1 be a discrete probability distribution
n
<Xl
all n, and r. c
o
n
= 1)
such that c
0
< 1.
(iii) Let the d.f. A(x) be thus defined:
A(x)
<Xl
= n=o
r
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
+co
(1.4)
f
IxlvM(lxl)dA(x)
<
is that both the following statements be true:
<Xl
(~.~.
c
n
> 0,
4
--I
+00
J
(1.5)
Ix IvM<l x I)dB 1 (x) <
I
00;
-00
(1. 6)
If. for the case v = 0. it should happen that (i).···.(iv) hold
except for the fact that
Ix n = +00
then one can still infer that (1.4) is
00
true if (1.5) is given, together with the condition El nc
n
If v and M(x) are as in Theorem 1 and if IT(s)
<
00.
°
= r~nsn,
<
s ~ 1,
is a probability generating function, then we shall say that IT is in the
class" (v, M) if
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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),
8(s), y(s) be probability generating functions,
8(s)
00
n
00
= Eobn
s , o
that rn
nb
<
00,
and only if both 8 and y belong
If, for the case v
= 0,
and that a(s)
to~(v,
°
~
s
~
= y(8(s».
1.
Suppose that
Then
a~(v,
~T
M) if
M).
00
it should happen that E1nbn
=
00
one can
still infer that a£'(O, M) if it is given that 8£1(0, M) and r7ncn <
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 £ > 0,
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2.
On
moment functions and some preliminary lemmas.
In Smith (1967) we have introduced a
A function M(x) belongs
(ii) M(x)
~
1, all x
~
to~
if:
(i) it is
0; (iii) M(x + y)
If M£2»and, additionally, (iv) M(2x)
*
M£~.
We say a d.f. F(x) is in the
~
class~
of moment-functions.
non~decreasing
in [0,00);
~
M(x)M(y) for all x, y
= O(M(x»
for all x
class~(v,
>
O.
0 then we say
M), for any real v
~
0,
i f ClxlvM( Ixl)dF(x)<W.From the point of view of defining a classb (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 ("ep(X) "" 1jJ(x) as x -+ 00" means that
there exist positive constants 0 and b such that
all large x).
o~(x) ~
1jJ(x)
~ b1jJ(x)
for
Thus we are led to introduce an extended class E1»* in such
*
N(x)£E~
a way that a function
N(x) x M(x) as x -+ 00.
*
i f we can find a function M(x)£.:tv such that
We can then say M(x) represents N(x)
Suppose that for some b
> 0
and some A
> 0
in~
*•
the function N(x)
satisfies the following:
(i)
(it)
(iii)
(iv)
N(x) is.non-decreasing for x
>
b;
N(b) > 0;
N(x + y) : AN(x)N(y) for all x
N(2x)
m
O(N(x»
>
b, y
>
b;
for all x > b.
Then we may suppose, with no loss of generality, that AN(b)
M(x)
= AN(x) ,
x > b,
= M(b) ,
x < b;
> 1
and define
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'*
to~.
the function M(x) so defined belongs
'*
(i), (ii), (iii) and (iv) above belongs to
Now suppose that, for some
> ~,
is non-increasing for all x
and define N (x)
v
~ ~
>
for all x >~.
= xVM(x)
above are satisfied by Nv(x).
Nv(X + y)
N (x)N (y)
v
v
=
E~.
2, M(x) is non-decreasing and M(x)/x
M(~)
and
Thus a function N(x) satisfying
O.
~
0 be a constant,
It is clear that (i) and (ii)
Further, if x
(x + y)v
v v
x y
Let v
• (x
+
~ ~
y)
M(x)M(y)
and y
~ ~,
{M(X + Y)
x + Y
}
But, since M(x)/x is non-increasing,
(x + )
y
f M(x++ Y)]
(x
y)
<
-
x
(I l)v { I
=: i' + y
I]
M(x) + M(y)
2
~ M(~) ,
since x
~ ~,
y
>~.
Thus N(x) satisfies (iii) above.
N (2x)
v
< 2v M(2x)
N (x)
v
M(x)
= 2(V+l)
< 2(v+l)
_X_{M(2X»)
M(x)
2x
-If.-f
M(x)
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x {M(X)j
so that
N)x + y)
N (x)N (y)
v
v
I
M(X)]
x
Finally, for x
> ~,
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* and
~
Thus N (x) also satisfies (iv) and so belongs to
v
function M (x) £'»* which represents N (x).
v
v
there will be a
We shall make frequent use of
these functions M (x).
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With these preliminaries set aside we can now prove the following.
Lemma 2.1
Let k > 1 be an integer and let M(x)
£)');1.
Suppose Xl' X2 , " ' , X
it
are independent and identically distributed random variables and write
Sj
= Xl +•..+
IM( IXII) <
Proof.
Xj for j
= 1,2,''',
k.
ThentM(ISkl) <
00
if and only i f
00.
Clearly M(ISkl)
~ M(t~IXjl), and
so, by a property of functions
in~
k
M(ISkl)
<
j~l M<!Xjl)·
Therefore
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and part of the lemma is proved.
If P{X I
trivial.
>
O}
= 1,
or if P{X I
~
O} = 1 the rest of the proof is
Suppose therefore that both P{X I
~
O} > 0 and P{XI
Then both P{Sk_1 ~ O} > 0 and P{Sk_1 < O} > O.
<
O} > O.
The finiteness of eM( ISki)
then implies that, for example,
is also finite.
However, if both Sk-l and ~ are negative then IXkl
<
ISkl
But ~ and Sk-l are independent.
<
O}
and so
must be finite.
Thus &{M(IXk!)IXk
8
is finite.
Similarly one can deal with &{M(I~I) IX
k
~ O}, and the lemma
follows.
Lemma 2.2
If, for large x, L(x) is non-increasing and xL(x) is non-
decreasing, then
lim
E.j..O+
L(l
lim sup
+
E
L(n)
n)
=
n~
=1
+ E n)
Iim i n f LeIL(n)
n~
Proof
If we set M(x) = xL(x), then
Mel
+
E
n)
> 1,
M(n)
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which implies that for all n
1 > L(l
1
I
+
E n) >
L(n)
1
- 1 +
E
The lemma follows at once from these inequalities.
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3.
Proof of Theorem 2
Let us write M (x), v
v
M (x) x xvM(x) as x
v
-+
0, for any member
>
of~
* such
Suppose that the integer v > 1.
00.
find a Kl > 0, K2 > 0, such that, for all large x,
(3.1)
For large R
>
0 and small
E >
0, define events
~I < E}
A (E) - {IS /n
n
n
{I Sn /n
B (R) _
n
=
I
~
-
M (IS
v
J
> R}.
n
I)
M (n~)
B (R)
dP.
v
n
Then
-tJ. (R, n) < Kl
v
-
K2n~
f
ISn I
B (R)
n
< -Kl
-
-
K2~
f
B (R)
n
Ixn I
Mv_1(lsnl)
dP
M _ (n~)
V 1
Mv_1(lsnl)
dP,
MV- 1 (n~)
that
Then we can
10
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by a defining property
f:,.
*
of~.
Hence we have
f
(R, n) < K1
-v
- K~ II
I I
(IS
n-1
(
)
nll
M
V-I
Mv ( X
n)
M
I)
dP
V-I
B (R)
n
1
>"2 nR
f
K1
2 1.I
+~
IXn -1.I1
>
t
nR
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(3.2)
Because X and S
n
n-1
are independent we have
fIx
(3.3)
n -lll
1
M
v
(IX 1 I>dP,
> 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, let1t
statements are true:
A
be the hypothesis that the following two
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~A(R,
(ii) For all sufficiently large R,
n)
-+
0,
as n
-+~.
The inequalities (3.2), (3.3) and (3.4) show that if 1M/ Ixll) is finite
and if~ (v-I) is true, then
(3.5)
~
for all large R.
v
(R,
n) -+
0,
as n
-+ ~,
Furthermore,
(3.6)
But, for some K4 > 0,
Mv (R + II n)
<
K4 (R + ll)
v
M(R + II n)_
M (lln)
v
(3.7)
<
since M(x)/x is non-increasing.
From (3.5), (3.6) and (3.7) we see that
1M (IS I)
v
n
Thus til A is true ifJt(A_I) is true and
show thatC1t'
o
Mv(lxll) <~.
Plainly we desire to
is true, from which it would follow (if &M (lxII> < 00) that
v
H v is true.
Let us write M(x)
of x.
Then
= xL(x),
where L(x) is a non-increasing function
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Is n IL( Is n I)
dP
nIlL(nll)
I Snn I dP.
J
< L(R - H n)
ilL (nil)
(3.8)
-
B (R)
n
Let us define, in a familiar way, for n = 1 2 ••••
"
X+. X if
'n
n
We then have X- = X+ - X ; Il+ = tx+ and
n
n
n
n
Il-
= IX-,
n
both necessarily finite because IIX I
n
and so on.
Il),
We then have (R »
J
\S:/ dP
=
B (R)
n
J
B (R)
,:n/ dP +
n
S+ > Sn - n
(3.9)
00; S+n = X+ + ••• + X+n'
<
-<
J
B (R)
Sn
s:
f
n
S+
n
>
S-
dP +
S+ > n(R+Il)
n -
S-
n
-
-+
J n(R-Il)
>
n -
By the weak law of large numbers, for any
P{IS+/n - Il+' < £}
/S:I dP
£ >
0
...!l dP > (Il + n
d
1, so that
S+
lim inf
n-+<x>
But &(S+ In)
n
= Il + ,
< £
and hence, for (R +
lim sup
n-+<x>
Il) > Il
f
S+ > nCR + Il)
n -
+
s:
+
£,
dP <
£.
...B. dP
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The arbitrariness of e tells us that
s+
...ll dP -+ 0,
n -+
n
00.
A similar argument will deal with the second integral in (3.9) and show
that
f ISnnl
dP
-+ 0,
n -+
00.
B (R)
n
This result coupled with the observation that L(R -
~
n)
< L(~n),
for R »
enables us to deduce from (3.8) that
f:::.
for R sufficiently large.
o
(R, n) -+ 0,
n -+
The proof that'M(IS
00,
n
I)
= O(M(n~»
now follows
easily from our inequality (3.6) with zero substituted for v.
Thus hypothesis(Jl
o
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
00.
We then have
f
M (R
<
+
v
B (R)n - A(e)
n
n
+
< K,.
~
by an argument we have already used.
~ n)
-v"'-M--::(:-n-:~)~- P{- An (e) }
(~
~
(V+l)
P{-A (e)}
n
Therefore, using the fact that
~,
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/). (R, n)
"
-+-
0 as n
-+-
00, we can infer that
&Is I""M( Is I)
n
n
(3.10)
=
J Isn I"M( Isn I) dP
+ 0 (M (nJ.l».
"
A (e:)
n
We first deduce from (3.10) that
(3.11)
< 1,
in view of the arbitrariness of e:
and Lemma 2.2.
We secondly deduce from
(3.10) that
tis I"M ( Is I)
(3.12)
lim inf _....:n::....._ _.;:.n_
n+oo
(nJ.l)" M(nJ.l)
>
-
--
J.I("+l) L(J.ln)
- 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.
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(ll - e:) (,,+l)l(u - e: n)
lim inf -n~
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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
<
00,
we can now appeal to Theorem 2 and deduce
that
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From this asymptotic relation and the assumption AE~(V, M) we are easily
led to the conclusion
The necessity part of the theorem then follows from the easily proved fact
that M(n].l) >< 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 = 0 and IX
n
the monotonicity of M(x),
=
00.
We have, by
16
But, since M(x)/x may be supposed non-increasing for all x
~
0, we have
inequalities like
and thus discover that
Mcl s
n
I)
I
< M( XII)
-
+- - -+
I I)-
M( X
n
Hence, if c! M( IXII) < 00, it follows that &M( ISn I) = 0 (n) and the theorem
is complete_
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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)
= Xl +...+
for their common d.f. and S
n
We suppose 0
<'
Xn , as
usual~
= P{X I
~
x}
for the partial sums.
Xl ~ +co •
The following Theorem A is an easy consequence of Theorem 6 of
Smith (1967).
In that paper it is supposed that eX I is finite whereas we
are here making no such assumption.
If IX I
truncation argument suggests itself readily.
= +00,
It will be interesting to see
how much is a consequence of this Theorem A.
Theorem A Let
~(x) £ »J
* and
If, for any constant y < &X I
then A(O) <
~
suppose
'
we write
and
f~ ~(lxl)dA(x) < ~.
If 'Xl
=~
we may take y arbitrarily large.
however, a suitable
18
We note that hex) is obviously a non-decreasing function of x
and that ~(Ixl) hex)
-+ 0
as x
-+
~
0
Thus, after an integration by parts,
-00.
the conclusion of Theorem A can be rewritten:
f~ hex) Id~(lxl) I
<
00.
This means
<
00
so that
Hny) -
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~(O)
'f - - - n- - n=l
Let M (x) represent x v M(x) in
v
enunciation of Theorem 1.
p{S
*,
~
< O} <
n -
00.
where M(x) is as defined in the
Then we deduce from Theorem A and (5.1) the
following:
Result 1
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and .!. fortiori
(5.1)
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(1.1) implies (1.2).
Elsewhere in Smith (1967) it is shown that the assumption eXl > 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}
<
00,
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••
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19
since M (2nn) )( M (n)
\I
as n
\I
Therefore
-+~.
But, for large n,
n
l
E M (2rn) > 2 (n - l)nM (nn).
r=l \I
\I
Hence
We have therefore established:
Result 2
(1.2) implies (1.1).
Let us now define, for n
piT < a}
n -
'IT
=
A
= n
n
n
= 1,2,···,
1
piS
<
n -
a}.
Then a familiar identity of Spitzer (1956) states that, for
a
<
s
From this identity, on comparing coefficients, it is clear that
'IT
1,
<
(5.2)
for all n.
Result 3
n
Thus we have:
(1.3) implies (1.2)
Finally, suppose again that (1.2) holds.
A
~
L A
n=l
n
<
~
Then we can set
>
An
20
and
o
l3(s)
<
s < 1,
will be a probability generating function in the class'(v, M).
Further-
more
y(s) = e
lI(s-l)
is the familiar Poisson probability generating function and y(s) is plainly
a member
of~(v,
M).
From Corollary 3.1 we can infer that y(l3(s»
is
another member of 1'(v, M) if only we can be sure that the "mean" of l3(s)
is finite,
~.~.
that
00
L nA <
n=l
n
00.
This is the same as the statement
r
n=l
(5.3)
When v
~
p{S
< O} <
n -
Result 4
~
1, we have
(1.2) implies (1.3)
However, we must also prove Result 4 when v
= O.
In this case
(5.3) is, in general, false. However, if we expand the Poisson generating
00
n
00
function y(s) = Lo cn s it is obviously true that Ll nC < 00. Thus the
n
case v
=0
_.
I
I
I
I
I
I
I
_I
00.
1, (5.3) is a consequence of (1.2), which we are assuming true.
Thus, when v
I
of Corollary 3.1 completes the proof of Result 4.
Theorem 1 follows from Results 1, 2, 3, and 4.
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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.
I
DCLlSSD'TIm
,
Spellrit,·
Classification
..
DOC UME NT CONTROL DATA· R&D
I.
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,
ORIGINATING
(If
abstract i1nd indt~xin~ annofiition must be entered when the overall report Is classified)
---
ACllVITY (Corporate author)
Za. REF"ORT SECURITY CLASSIFICA.TION
Zb. GROUP
,
r'
R F.: P 0 R TTl
1;-
Rr.S1JLTS
SOME
4.
DESCRIPTIVE NOTES
usna
GDEltAL MOMI:II'.r J'WCIrIOIS
(Type of report and inclusive dates)
Technical leport
5
AU THORIS)
(First name, middle initial, last name)
Smith, Walter Laws
tj.
7a. TOTAL NO. OF PAGES
REPORT DATE
17b.
NO. 0 F RE FS
.O'Y8a'ber 1961
8a.
CONTRACT OR GRAt>JT NO.
h. PF<O.JEC T NO.
9a. ORIGINATOR'S REPORT NUMBERlS)
.60-855-(09)
RR-003-05-Ol
IIlstit.t. of Statistics E_o Series
bm.er 55~
c.
9b. OTHER REPORT NO(S) (Any other numbers that may be assigned
thIS report)
,t.
I-;-- ':'T~I:"'UTIOf'J
STATEMENT
Distriltution of the doCWl8Dt is unlimi. ted
,,
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'Sf'crJrit\' c!n,c.."ificatirJn of fit/(!, ho(!):
_~
IJ P P L EMF f'J T A
.~
y
"J 0 T E 5
12. SPONSORING MILITARY ACTIVITY
J.ogistiCI and Jathematical Statistics
Branch Oftice of I&val Research
WUhiqtoa, D. c. 20360
1'
APST[:~ACT
Let
(~)
'be a sequence of independeDt and identicall7 distributed random
variable. lIuch that 0 <
be an integer and let
M(x)
M(x)/x is DOD-increasing
and write Sn .~ +12 +••• + Xn Let ",>0
be a IIOD-decreuiDg tu.nctioD ot x ~ 0 such that
~ ==
eXD<+
aDd
M(O)
>
-
00
o. Then it ellll"'M(lxll) < 00 ad Jil<
00:
it
follows that e Isn I"'M( Isn I> - (nlil) "')(n~) &8 n - . 110. It pallO (v .. 0) then
eM(lsDI> =0 (n). A variet7 of results stem from this ain theorem (neorem 2),
concerning a closure propert7 of probabilit7 generating functions and a random.
walk result (!heorem 1) comaected wi th quee•.
I
Sccuritv Classification
I
.• '-j
Security
..
cl•••ific.tion
14.
~'NK
K KY WO"D'
"O~K
~'NK
•
WT
"O~K
~INK
•
WT
lleneval lfheo17
lladom Walks
RO.~
I:
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WT
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Ibaents
Pro'babilit7 GeneratiDg hJlctiOJlS
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••
Security Classification
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