PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 50, July 1975
RECORD TIMES
J. GALAMBOS AND E. SENETA
ABSTRACT. Let X., X 2, ••■be
random
variables
a record
time
with
The
strictly
increasing
vestigate
2.
the
We use
smallest
1 = L(l)
sequence
sequence
valued
such
that
limit
1. The results.
(i.i.d.)
random
Fix).
We define
laws
than
< L(3)
distributed.
for
U(n)
This
U(n)=
fact
the functions
the assumption
of L(t2) does
determined
of continuity
of Fix).
on Fix).
by considering
This
immediately
eous
Markov
chain
yields
continuous
(D
(2)
the uniform
and the conditional
k.
Received
This
function
and are well defined
to see
that
the distribution
distribution
L(3),
the distribution
of Lin)
Fix)
L(2),
by
can be
= x for 0 < x < 1.
• • ■ forms
a homogen-
distribution
;>2,
probabilities
PiLin) = k\Lin-
of 7 and
distributed
distribution
variables
PU(2) = 7) = 1/7(7 - 1).
and with transition
to de-
72> 2.
It is easy
Hence,
that the sequence
with initial
it possible
Lin) by L(l) = 1 and
L(t2) are random
not depend
as the
turn out to be
and identically
L(«) = min{/: X.> XL, _1)},
Evidently,
defined
makes
we in-
— 1), zz >
and for A(zz) = L(zz) — L(n — 1), zz > 2.
with a common
the record times
U(n),
is a
note
L(n)/L(n
approximations
Let X„ X2, • • • be independent
variables
and we put
times
In the present
T(zi) to
These
X.,
< • • • of record
the ratios
< T(n).
distributed
zL(zz) Ï ? is called
any previous
variables.
approximation
and identically
several
larger
through
ü(n)
function.
< ¿(2)
of random
ÍL(zz)i
an integer
integer
independent
duce
sequence
and identically
distribution
if X. . , is strictly
L(l)=l.
independent
a continuous
l) = 7) = j/kik - l)
probabilities
elementary
by the editors
AMS (MOS) subject
above
fact
November
classifications
for k > 7 + 1 > « > 3,
are equal
was first
recognized
15, 1973 and,
(1970).
to zero
in revised
for any other
by Rényi
form,
[3],
March
Primary 60F05; Secondary
values
and he
18, 1974.
60JI0,
62G30.
times,
Key words and phrases.
limit laws, maximum.
Continuous
distribution,
record
times,
ratios
of record
Copyright © 1975, American Mathematical
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383
Society
384
J. GALAMBOS AND E. SENETA
himself
done
deduced
several
by several
most recent
survey
Uin),
number
called
- l), 72> 2.
were guided
theory.
The fact
Engel's
Of course,
we investigate
the following
our investigation
series,
and their
research
on this
has
since
topic,
references,
see
been
the
or, alternative-
[7].
paper,
(i(72) = Lin)/Lin
further
Out of the papers
[4], [5] and [6, p. 8l],
In the present
sequence
consequences;
mathematicians.
ones
ly, the forthcoming
tios
of its
that
has
we do not need
the sequence
Though
several
Lin)
results
through
are known
the rafor the
statements
appear
to be new.
by a recent
result
[2] by one of us in metric
record
been
times
noted
this
are related
earlier
relation
to a series
by Renyi
and we shall
Our results
and
expansion,
[3] and Vervaat
make no further
[6].
reference
to it.
We formulate
Theorem.
ities
our main result
Let the integer
Tin) - 1 < Uin) < Tin),
as a theorem.
valued
n > 2.
function
Then
Tin)
the random
be defined
by the inequal-
variables
Tin)
are
i.i.d.
with
PiTin) = 7) = 1/7(7 - 1).
Several
duce
that,
consequences
as
of our theorem
7 > 2.
are worth recording.
(3)
(4)
and that,
for any real
lim P(maxÍL/(2),
In another
direction,
z > 0,
(7(3), •••,
our theorem
Uin)\ < nz) = exp(can be applied
Mn) = Lin) - Lin - D,
with
Lin-
l).
Indeed,
(5)
several
observing
that,
ÍA(72)>sL(72-l)¡
results
of integers
that,
de-
[U(2) + (7(3) + • • • + Uin)]/n log 72-» 1
in probability,
holds,
We shall
T2—»+ 00,
are immediate.
2 < n < N for which
the Theorem
l/z).
to compare
n > 2,
for any positive
= ¡T(72)>5+
integer
if a(zV, s) denotes
the inequality
on the left-hand
of the strong
s,
lj,
For example,
and (5), in view
the gaps
law of large
the number
side
of (5)
numbers,
imply
as N—• + <xt
J..
aiN, s) =- 1 VI = 1
1.
P I lim-—
y
In particular,
the case
s = 1 yields
n
s +1)
that,
almost
surely,
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the number of integers
385
RECORD TIMES
n < N fot which
Ain) > L(?2 - l)
Ain) < L(t2 - l).
rem, further
By the central
information
theorem
equal
sin)
of 72 leads
it was not our aim to reobtain
to those
and the iterated
on a(/V, s) is immediate.
of 5 in (5) as a function
However,
is approximately
limit
theo-
On the other hand, the choice
to previously
known
for which
logarithm
known
results,
hence
limit
results.
we do not give
further details.
In concluding
placed
by a strong
this
section,
limit law,
we remark
as we shall
that
see
the limit
in (3) can not be re-
at the end of the next
section.
2. Proofs.
Proof
of the Theorem.
{Tin) > 7! is equivalent
Notice
to \Uin)
that,
by the definition
> 7! for any integer
of T(t2),
7 > 1.
the event
Thus
P(T(2)>;2, T(3)>/3, •■•, Tin)yjJ
(6)
= P(Í7(2) > y2, 1/(3) > f y
",
U(n) > j)
= £ P(L(2) = k2, L(3) = ky • • •, Un) = kj,
n
where
¿n
is summation
over
all
(72 - l)-vectors
ik-,
k,,
k2 y j2 and kt y f,k._ j for 3 < t < 72. We first observe
• ■ • , k ) for which
that
£ PÍLÍ2) =k2, L(3) = hy • • •, L{n) = k)
n
(7)
= £
n-
On the other
(putting
P(L(2) = k2, L(3) = ky ■• -, L(n - l) = kn_ p L(t2) > fnkn_ {).
1
hand,
by the
2* for summation
L's
forming
over
a Markov
r with
7 k
P(L(2) = k ¿ L(3) = k-,
...,L(n-l)
5
chain
and by (2) we have
, < r < + 00)
= kn— „L(n)yjk
1
= P(L(2) = ¿., L(3) = Ä„ ••-, L(»-l)
'n
A
n—1
AY* _JLlL
Kr-1)
= H/Jn)PiLi2) = k2, L(3) = k}, - •., L(n - l) = ^_ j).
This,
in view of (6) and (7), gives
P(T(2)>/2,
r(3)>/3>
=*
that
•••,
Tin)>jn)
~ (l/fn)P(T(2) y j2, • • •, Tin - 1) > /„_,),
and thus
by induction
we get that
P(T(2) > 72, T(3) > 73, • • •, Tin) > jj
= l/(/273 ■■' /„)•
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386
J. GALAMBOS AND E. SENETA
The above
pendent
equation
implies
that
with distribution
the random
variables
T(s),
s > 2, are inde-
P(T(s) > /) = 1//, 7 > 1. Therefore
P(T(s) = 7) = P(T(s)y 7 - 1) - P(T(s) y j) = l//(/ - D
for any integers
In order
s > 2 and
to prove
7 > 2.
The Theorem
the relations
is thus
established.
in (3) and (A), we note
that,
by definition,
for all s > 2, 0 < T(s) - U(s) < 1. Therefore,
(8)
n
n
0< X
T(s) - £
s =2
Uis) < 72
s=2
and
(9)
0 < max{T(s):
Now,
since,
as
2< •><72| - max{(7(.s):
2 <s
<n\
<1
72—>+ 00,
T(2) + T(3) +:••• + T(tz) ~ tz log 72
in probability,
istic
which
functions,
easily
and,
also
P(max|T(s):
referred
in this
(3) and (A).
theory
concrete
The fact
vergence
yield
to existing
simple
case,
dicts
from the result
which
the general
result
actually
express
(In the above
laws
proof
random
we could
variables,
but
is very easy.)
then (8) would
with infinite
argument,
for i.i.d.
by an almost
As a matter
imply
expectation.
a similar
This
sure
con-
of fact,
limit
however
if
law
contra-
of [l].
(8) and (9), further
the fact
turn to the independent
< 72Í, we have that,
to character-
< nz) ~ exp(-l/z),
of [l] and from (8).
one,
are i.i.d.
From the inequalities
They
by turning
that the limit in (3) can not be replaced
follows
T's,
of limit
a direct
(3) were true with probability
for the
from the Theorem
2 < s < 72! < nz) = Pn~1(T(s)
(8) and (9) immediately
have
follows
by our Theorem,
T's.
that,
in order
For example,
with probability
one,
limit
laws
can be deduced.
to investigate
putting
the
U's,
U*(n) = max j (/(/):
we can
2 < 7
log U*(n) ~ log 72 as T2—>+°°.
REFERENCES
1.
Y. S. Chow
and H. Robbins,
infinite moments and "fair"
games,
Ozz sums
of independent
random
variables
with
Proc. Nat. Acad. Sei. U.S.A. 47 (1961), 330—335.
MR 23 #A29082.
J. Galambos,
Further
ergodic
results
on the Oppenheim
series,
Quart.
J. Math.
Oxford Ser. 25 ( 1974), 135- 141.
3. A. Renyi,
Theorie
du Colloq.
de Math. Renyi
Biaise
Pascal,Tome
des elements
saillants
a Clermont
a l'Occasion
d'une suite d'observations,
Actes
du Tricentenaire
de la Mort de
II, Ann. Fac. Sei. Univ. Clermont-Ferrand
MR 44 #3376.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
No. 8 (1962),
7—13.
RECORD TIMES
4. S. I. Resnick,
5.
R. W. Shorrock,
Record
values
Ozz record
and maxima,
values
and record
387
Ann. Probability
times,
J. Appl.
1 (1973), 650—662.
Probability
9
(1972), 316-326.
6. W. Vervaat,
Success
epochs
in Bernoulli
trials with applications
in number
theory,
Mathematisch
Centrum
Tracts
42, Amsterdam,
1972.
7. —-—,
Limit theorems
for partial
maxima and records
(to appear).
DEPARTMENT OF MATHEMATICS, TEMPLE UNIVERSITY, PHILADELPHIA,
PENN-
SYLVANIA 19121
STATISTICS DEPARTMENT, S.G.S., THE AUSTRALIAN NATIONAL UNIVERSITY,
BOX 4, P. O., CANBERRA 2600, AUSTRALIA
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