*Currently on leave at Cambridge University, England
.~
•
tResearch sl1Pported by O:f:f~ce9( Na.va.l_Re£ea_rch undercontract-N00014-67~­
-- A-032l-002, Tsk NR-042-214.
LOCATING THE MAXIMUM OF A STATION.L\RY NORMAL PROCESS t
by
M.R. LEADBETTER*
DepaPtment of Statistias
UniveX'sity of NOX'th CaX'oZina~ ChapeZ
HiZZ~
Institute of Statistics Mimeo Series
No~
August, 1973
N. c.
887
LOCATING THE MAXIMUM
OF A STATIONARY NORMAL PROCESSt
by
M.R. Leadbetter*
Department of Statistics
University of North Carolina, Chapel Hill,
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1.
N.C~
INTRODUCTION.
Considerable attention has been paid in recent years to questions surrounding the maximum value M(O,T)
the time interval
0
~
t
~
T.
of a stationary normal process
~(t)
in
In particular the double exponential asymptotic
- bT) ~ xl (as T ~~) has been obtained under avariety of conditions, (e.g. (Cram~r 1965) (Pickands 1969) (Berman 1971)). (The
form of
Pr~T(M(O,T)
exact distribution of M(O,T)
for finite
T is unobtainable in general).
A problem suggested to me by D.R. Cox concerns the position L
say, of that point of
[O,T]
= LT,
where this maximum is attained, and, in parti-
cular, whether the distribution of L is uniform over this range.
In this
paper we show that the distribution of L is not, in general, uniform for
finite values of T (and indeed will "usually" have jumps at 0, T,
that it is asymptotically uniform as T ~
but
~.
In section 2 we consider some aspects of the corresponding problem in
discrete time, which give some feeling for the main results.
Section 3 con-
tains a discussion of the' -presence of jumps, and the sYmmetry of the distri-
*Currently on leave at Cambridge University, England.
tResearch supported by Office of Naval Research under contract N00014-67
A-032l-002,-TSK NR-042-2l4
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2
.~
bution of LT for finite T. In section 4 asymptotic uniformity of Lr is
shown. This relies on a bivariate form of the asymptotic distribution of
M(O,T) which - along with further aspects of this work - will be discussed
in (Leadbetter 1973).
2.
Let
Mn
{~:
n
= max(t l
n
= 0,
DISCRETE TIME CASE.
± I ••. } be a discete parameter stochastic process arid
Let
t2
Ln
be the "position of the maximum" i.e. a
if t k =Mn • (To avoid
ties, it will be convenient to assume that the joint distribution of ~l •••
random variable taking the value k (1
S
k s n)
is absolutely continuous.)
If the
~i
are independent and identically distributed, it is immediate
Ln is uniformly distributed on the itegers 1 .•• n, Pr(Ln =k) =-nI .
That is, any of the t k is equally likely to be the largest of t l •.• t n
that
A small opinion survey shows that it would be reasonable to conjecture that
the same would be true if the
were not independent, but a stationary
~i
sequence.
This conjecture is, in fact, false.
For example suppose
standard normal random variables such that
and zero otherwise, where
~ts
s~.
=P
for
J
are all
li-jl = 1
(This may be achieved by forming the
as "two-term moving averages" of a sequence of independent random vari-
abIes.)
Consider the simplest case,
Pr(L 3
-e
Ipl
cov(ti,t j )
t.
= 1)
n
= 3.
=
Pr(t 1
~ ~2'
=
Pr(n l
~
Then
~
tl
0, n2
~
t 3)
0)
where n1 = (~1-~2)/[var(~1-~2)]~' n2 = (tl-~3)/[var(~I-t3)]~ are standard
normal random variables with correlation Al = ~(I-p)-~. It is well known
~n
3
'e
Pr(L 3
.Similarly Pr(L 3
= 1) = ~
1
.-1
+ 2n S1n
AI'
= 3) = Pr(L3 = 1) , whereas
1
-1
Pr(L3 = 2) = %+ 2~ sin A2
A2 = (1-2p)/[2(1-p)]. If p > 0, it follows that Al > A2 and
P(L3 = 1) = P(L3 = 3) > P(L3 = 2). (For p < 0 the inequality is reversed.)
Thus even in the case n = 3, the position of the maximum is not uniform,
where
even though the random variables concerned belong to a stationary sequence.
One can see an intuitive reason for this.
example, with general
n.
For consider the above
Then in "assessing the chances" that
;1 will be
the largest, we notice that it is correlated with only one of its competitors
On the other hand ;2 ••• ;n-l are each correlated with two compe-
(;2)'
titors.
Of course the structure is more complicated than this since we are
dealing with multidimensional, and not just bivariate distributions, but this
makes it even more likely that the situation will change as we move away from
the edges of the sequence.
Thus we may call this an "edge effect" though it
does not necessarily disappear for
;.1
near the middle of the sequence,
especially if the correlation structure dies off slowly.
On the other hand
it does suggest that we may (a) have uniformity for
Ln under a "circular"
stationarity assumption and (b) in any case we may have asymptotic uniformity
as n becomes large.
We consider the first of these conjectures now, and
the second (or rather its continuous time version) in section
Specifically let
covg.
·e
1
~.)
J
=. A...
1J
~l
~l'"
4"
•••
~n
~
will be called "circularly" stationary if there
~
be standard normal random variables with
4
exists r k , 0 s k s n - 1 such that
(a)
un
(b)
lSi, j S n
A." = r C.'1-3
.')
_lJ
r
= r _
1 S k S n - 1
k
n k
This concept has been used for example in~annan 1960).
(a) is the usual weak stationarity requirement.
r n _l
=rl
,
so that we may regard
r 1 as if "n" and
/;n
and
~l
"1" were adjacent points.
tionarity when we regard the points
1 ••• n
Condition
(b) gives, in particular,
as having the same correlation
In other words we have sta-
as arranged on a circle.
Theorem 2.1. For circularly stationary normal random variables
~l
...
1, 2
~n
, the position Ln of the maximum is uniformly distributed on
... n •
Proof. Let f(xl ••• xn ) denote the joint density of l;l ••• I;n' From
the circular stationarity it may be shown without difficulty, that, for all
f(x l , x2
xn) = f(xn , Xl' x2 ••• xn _1)
The joint density g(x1 , x2 ' •.• x ) of the random variables
n
(I;i' 1;3 ••• , I;n' 1;1) is ':f(xn , xl'· x2 ••• Xn _l )· Hence writing
(2.1)
A = {(Xl' x2 ••• xn ) : Xi S Xl i = 2•.• nl we have
Pr{L=2} = I g(x1 , x2 ••. xn) dx l
A
by (2.1).
But this is just Pr{L=l},
Pr{L=I} = Pr{L=2} =
3.
dXn
and,proceeding in this way, we have
=!n
CONTINUOUS CASE.
JUMPS AT
0, T
AND SYMMETRY.
We consider now a (zero mean) stationary normal process
I;(t)
in con-
tinuous time, with zero, mean, unit variance, and having covariance function
5
ret)
such that
val - [O,t]
A
2
atwhich
= -r" (0) \
~(t)-
<
Let
ClO.
L.r denote the point of the inter-
(first)- attains its maximum in
is not, in general, uniformly distributed over
[O,T].
that~ange.
L
T
However a simple
case in which this can occur is that of a "periodic process"
= A cos(wt
~(t)
where A,
are independent random variables,
~
bution and
(t(t)
being uniform over
~
(-n,n),
is a stationary normal process.)
(2n/w)
of this waveform,
and is uniform over
L
(O,T)
L.r
=0
or T,
A having a Rayleigh distri-
and w is a fixed constant.
As long as
T is at least a period
can be regarded as uniform over
T
if T
= 2n/w
On the other hand if T. <2n/w,
that
+ ~)
(O,2n/w),
•
then there is positive probability
viz
Pr(L.r
= 0) = Pr(LT = T) = ~(l
wT,
- 2n1
and
(0 < x < T)
Thus the distribution of
L,
consists (for T
<
2n/w)
of equal jumps
at 0, T and a uniform part in 0 < x < T •
The existence of jumps of the distribution of L at
due to the special nature of this example.
0 and T is not
Indeed suppose that
~(t)
is
any stationary normal process with zero mean, and whose covariance function
is four times differentiable at the origin.
ings of zero by the derivative
so that
EN < 1.
~'(t)
·e
where T is chosen
in (O,T) or ~'(t)<O in (O,T)} = Pr{N = O}
~
Pr(L
in 0 S t S T,
Then
Pr{~'(t»O
and hence
Let N be the number of cross-
= 0)
~ Pr{~'(t)
1 - EN
>
0
< 0 in (O,T)} > 0
(since by sYmmetry this
6
also equals
u
Pr{t'(t) > 0 in (O,T)}) • Thus' the distribution of L . has
a jump at zero.
A stationary normal process
net) = t(-t)
Ht)
is "reversible" in the sense that
has the same finite dimensional distributions as
property leads to symmetry of the distribution of L in (O,T)
Pr(L s x)
= Pr(L
~
T - x»
t(t).
This
(i.e.
and, in particular, the fact that the distri-
bution of L has the same jump at
T as it does at zero.
This will be
developed further in (Leadbetter 1973).
ASYMPTOTIC UNIFORMITY OF
4.
~.
In this section we again assume the conditions stated at the start of
section 3 hold.
AT(O<A<l)
Since the maximum of t(t)
if and only if M(O,AT)
~
in 0 s
M(AT,T),
t
S T occurs by time
Lr
the distribution of
is
given by
Pr{LT SAT} = Pr{M(O,AT) ~ M(AT,T)}
The asymptotic behaviour of this distribution is considered in the following
result
Theorem 4.1. For the stationary normal process considered, if
provided either
(i)
Pr{L S AT}
T
ret) log t ~ 0
~ A
as T
~ ~
2
o r (t) dt < w
is "asymptotically uniform" over [O,T].
this precise sense
or
(ii)
0 < A< 1
f
~
Thus, in
LT
Proof.. It is known (Berman 1971) that if either (i) or (ii) holds
(in addition to
A2 <
~
assumed) then
Pr{~[M(O,T) - bTl s x} ~ exp(_e- x ) as T ~ ~
where
~
·e
= (2 log T)%
b = (2 log T)~ + log (A %/2n) / (2 log T)%
2
T
7
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It may be further shown, under the same conditions (Leadbetter 1973)
that Pr{X S x, Y S y} ... exp (-~ -x) exp (-e -y) as T ... CXl where
T
T
XT = aAT[M(O,AT) - bAT] YT = apT[M(AT,T) -b pT ] (p = 1 - A) • Thus
aAT
Pr{LT SAT} = Pr{Xr - apT YT ~ aAT(b pT - bAT)}
Now aAT/apT '" 1 as T'" CXl and it is easily checked that
aAT(b pT - bAT) ... log(p/A) • An obvious weak convergence argument (the
:
details of which will appear in somewhat more general terms in (Leadbetter
1973) then shows that
Pr{L SAT} ... Pr{X - Y ~ log(p/A)}
T
where X and Yare independent random variables with common distribution
function exp(-e -x ) • This can be written as
by integrating over the appropriate part of the plane, and this reduces, by
some amusing algebra, to the required value
A.
Finally we note that several authors have shown that the asymptotic
distribution of M(O,T)
holds in certain circumstances even if A2
=
CXl
(e.g. Pickands 1969).
The implications of this in the present context will be taken up in
(Leadbetter 1973).
SUMMARY
The purpose of this paper is to describe certain results concerning the
location LT of the maximum of a stationary normal process in the time
o S t ST. It is shown that LTdoes not, in general, have a uniform distri-
.e
bution on 0 S t S T,
T ...
CXl.
but is asymptotically so under weak conditions as
The various features of the corresponding discete time problem are
also discussed.
8
REFERENCES
Berman, S.M. (1971)u "Asymptoticindependence of the numbers· of high and low
level crossings of a stationary Gaussian process,ll AnnaZs Math. Statist.
42.. p. 927.
Crdmer, H. (1965) "A Limit theorem for the maximum values of certain stochastic processes," Teor. Ver. i Prim., 10, p. 137.
Hannan, E.J. (1960) "Time Series Analysis" Methuen
&Co.,
London.
Leadbetter, M.R. (1973) liOn joint distributions of stochastic process extremes,
wi th implications for their location", in preparation.
Pickands, J., III, (1969) "Asymptotic properties of the maximum in a stationary Gaussian process,ll Trans. Amer. Math. Soc., 145,: p. 75.
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