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LIBRARY
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ALFRED
P.
SLOAN SCHOOL OF MANAGEMENT
A CONVERGENCE THEOREM FOR EXTREME VALUES
FROM GAUSSIAN SEQUENCES*
-by
Roy
E.
Weisch
Working Paper 550-71
June
1971
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS
02139
Imass. wst. tech.
AUG
2
1371]
OEWEYLlSRARt
A CONVERGENCE THEOREM FOR EXTREME VALUES
FROM GAUSSIAN SEQUENCES*
Roy E. Welsch
Working Paper 550-71
June
1971
*This research was supported in part by the U. S. Army Research
Office (Durham) under Contract No. DA-31-124-AR0-D-209.
no. ^50-7
RECEIVED
AUG 13
M.
I.
T.
1971
LlbRAHitS
MOS Classification Numbers (1970):
Primary
62G30
Secondary
60G15, 62E20
Key Words:
order statistics, Gaussian processes, extreme-value
theory, weak convergence.
537:09
Abstract
Let {X
,
n=0, ±1, ±2,...} be a stationary Gaussian stochastic process
with means zero, variances one, and covariance sequence {r
max {X, ,...,X
}
=
and S
second largest {X, ,...,X^}.
obtained for the joint law of M
and S
limit law which is
a
a
function of
of dependence provided either
a
Let M
=
Limit oroperties are
as n approaches infinity.
A joint
double exponential law is known to hold
if the random variables X. are mutually independent.
sidered Berman has shown that
}.
When M
alone is con-
double exponential law holds in the case
r log n
or
^-
r
I
<
«.
In the
present
work it is shown that the above conditions are also sufficient for the
convergence of the joint law of M
stochastic processes Mr
^i
and Sr
and
^-,
S
with
.
Weak converaence nronerties of the
<
a
<^
t
<
« are also discussed.
1.
Introduction
This paper extends and simplifies a theorem obtained by
.
the author in section 4 of [6].
The reader is assumed to have some
acquaintance with those results.
Let {X
n=0, ±1, ±2...} be a discrete parameter stationary Gaussian
,
stochastic orocess, characterized by expectation, and covariance function,
respectively:
E(X^) ^ 0,
(1.1)
E(^ h.n^'=\''Q--^
This paper treats some of the limit properties of the random variables
M^ = max {X^ ,..
S
.
,X^},
second largest {X, ,...,X
=
A double exponential
variables
X.
}.
limit law is knov/n to hold for M
are mutually independent, that is r
if the random
i 0, n / 0.
Berman [2]
has shown that the same law holds in the case of dependence provided either
(1.2)
(1.3)
loq n
r
I
r^
<
-+
0, or
«..
n=l
The author [8] has shown that the processes {Mr
.
-i
,
Sr
.-,}
,
prooerly
< a <_ t < «, converge weakly
withO<a<_t<«>,
wee
in the Skorohod snace D
normalized, and with
when the Gaussian sequence is strong-mixing and
2
[a,°°]
r^ log n = 0(1),
(1.4)
The limit law is the same as that which occurs in the independent case.
Condition (1.4) is weaker than (1.2) but we imposed the strong-mixing
condition.
natural
In
many cases strong-mixing is difficult to verify and it is
in view of Berman's work to see if the weak convergence results
mentioned above hold when the strong-mixing assumotion is dropped and
just (1.2) or (1.3) is assumed.
this is, in fact, true.
The purpose of this oaper is to show that
The reader is referred to [7] for some examples
of why it is of interest to consider the joint distribution of M
n
and S„.
n
A more extensive discussion of the maxima of stationary Gaussian processes
is contained in [4].
Some Properties of Gaussian Distributions
2.
Let (r..) be a k x
.
k
symmetric positive definite matrix with 1's along the diagonal, and let
()),
(x, ,.
.
.
,x.
r..,
;
<^i
1
<
j
£
k) be the k-dimensional
Gaussian density
and covariance matrix (r. .);
function with mean vector
1
of the x's and the k(k-l)/2 parameters
r.
..
J
i>.
K
is a function
Define:
Q^(c,d,a,{r.j})
(2.1)
•••
/ dx^
/ dx^_i
dx^
/
Q
—CO
-00
/
••.
dx^^^
/
from d to « will always be on the a
The integral
assume that
<
dx^
•
*^(x^
.
.
.
,x^;{r
})
'-'
dummy variable and we
The oartial derivative with respect to
< d.
c —
,
— oo
—00
r,
„
is ob-
hi!,
tained by the method of Slepian [5]:
3Q.
(2.2)
h^
~
c
= /
...
c
•••
/
/
<t)|^(x^ ,.
.
,Xp^_i
,c,x^^1
,.
.
,x^_pC,Xj^^1
,.
.
,X|^)
d
when
(2.3)
h
^ a,
a
If
dXj
3^h
^ a,
h
^
I,
and
J
*^(x^
"
aSi
/
-oo
.
.
.
— oo
,.
.
.
.x^_^ .d,x^^^
,.
.
.
,x^.^ ,c,x^^^ ....
,x^)TT dx
.
with
a
corresponding exoression when
/ a
h
of integration in (2.2) are reolaced by
integral
is
increased.
and
a
(«',..,«')
= a.
If the UDoer limits
then the value of the
Now integrate k-3 variables from -« to +« to obtain
!^(c,d,a,{r..})
(2.4)
^
I (J)-(c,C,X
:
z(h,£,a))dx
where
ha
h£
E(h,il,a) ^
la
ih
al
ah
(2.3) are reolaced by
We note that if the limits of integration in
(<»,.
.
.
then
,0°)
9Ql
(2.5)
<_
9r
<|,2(c,
c;
r^^) = (2^)"''
(l-r^^)-W[-c2/(Ur^^)]
ail
Since {X
j
-
i,
i
stationary process,
}
is a
<
j; we write r.
.
=
r.
..
r.
.
is a
function of the difference
The function
P.
is defined as
P,^(c,d,a,r^ ,...,r,^_^) = Q|^(c,d,a,{r.j}):
the partial
derivatives are given by the chain rale as
{a„} and {b„} be defined as
Let the sequences
^
n
a^ = (2 log n)
n
h
(2.6)
"
p
^^
^°^ "^^
"
^^^ ^°^ n)'''^(log log n + log
4tt),
'^n
It is known
lim P{M
n^^o
[3]) that when r^ = 0, n
(cf.
n
—<
X + b
a
n
}
M
exp (-e"^) k G(x)
=
n
for all x.
Both (1.2) and (1.3) imply that r
positive number
6
such that
=
6
<
sup
|r
1
+ 0; therefore, there exists
a
1.
n
Define:
6(n) = sup
k>n
|r.
],
=
q
"
"
^
lim 6(n) log
n
= 0,
=
0.
n-x"
(2.8)
lim 6^ log
n^^
n
"
.
Clearly (1.2) implies that
(2.7)
n^, 6^ = 6(q„/2) where
and
<
B
<
(1-6)^/2(1+6),
Convergence Theorems
3.
and S
the joint laws of M
Theorem
1
Let {X
.
section we extend Berman's results to
In this
.
.
n=0, ±1, ±2,...} be a stationary Gaussian sequence
,
If either
satisfying (1.1).
lim r
log n =
n
or
r^
y
<
00
n=l
then
lim P{M
n—<
a X + b
n
nn—
<
S
,
a
y + b
n-^
}
n
G(y){l + log [G(x)/G(y)]}
y
<
x
G(x)
V
>_
X.
The following three lemmas will be needed in the oroof.
let c
=
<
e
n
n
assume that
Lemma
+ b
V
a
n
1
<_
and d
=
x + b
n
is so large that c
n
n
>
,
and to avoid technical details we will
0.
Assume that the conditions of Theorem
.
1
Y
1
Then
•
_
(3.1)
a
n
1
-
<i>(b
Y
)
=
0(n
2
")
For convenience
1
are satisfied and
10
where $(•) is the standardized Gaussian distribution function.
Proof
use the following approximation for the tail of the
We shall
.
standard normal distribution ([1], page 933):
-1-
(3.2)
-
1
by
n n
Since
$(x)
-»-
1
(2TT)"^;^exp
<
x > 2.2
/2)
« we can ignore
the constraint x
^
>
2.2.
(3.1) we note that
To prove
b^ = 2 loa n
(3.3)
{-y.
X
-
log log n + 0(1)
and then use (3.2) to obtain for large n
1
-
o(b^Yj ±
K'n
The conditions on y
'n
Lemma 2.
Y^ =
"
"/Yn((1og n)
now imoly
J that (3.1)I holds for all
\
If the conditions of Theorem
(1
-
o(l)).
-
26^
3(5^/(1 +
1
hold and
then
n-1
(3.4)
lim n
<},
(c
,c
,6
)
\r.\
I
i=q +1
n->«>
=
^
and
(3.5)
lim
n-oo
P
n'^Cl
n-1
-
$(b y )]<|)o(c
n^n'-"^2^
,c
,6
)
n' n' n'
T
Ir.l
j'
i
^^^
^^
= 0.
n,
11
Proof
The oroof of (3.4) is contained in Theorem 3.1 of the paper by
.
We consider (3.5) first when
Berman [2].
(1.2) is assumed.
Lemma
1
implies
that
1-y'
n[l
-
Hb^y^n iKn
".
Now
1
-
Y^ =6 {(10
'n
56
-
n
n
)/(l
+
)^} =
2<S
6
n
n
0(1)
'
and using (2.8) we have
n
^-n"
=
0(1)
which, when combined with (3.4), yields (3.5).
When (1.3) holds, the Schwarz inequality implies that (3.5) is less
than
(3.6)
2rn
n^[l
-
./u
Ni2,2,
.(b^vjn^(c^,c^,5jn^
.
Finally we use Lemma
1
3
,
";^
2
J
and substitute for
r^
*2^^n'^n''^n^
(3.6) is dominated by
-4/(1+6
3+6 0(1)
„
r(log2 n)n
"
•
n
)
n-1
""
I
2
r
j=q +1
which tends to
0,
completing the proof of Lemma
2,
^° ^^^^
^"*^
'*'n
12
Lemma
If the conditions of Theorem
3.
(3.7)
lP^(c^,d^,a,r^,...,r^_^)
I
-
1
hold then
P^(c^,d^,a,rp.
.
.
,0,...,0)1 ^
,r
a=l
Proof
i
= q
.
^n
n
By the law of the mean, there exist numbers r!
+1 ,.
.
.
,n-l
J
and r.
such that
,
Pn(S'^n'°''^l"--''n-l)
-
between
,1
'
Pfi^^n'^n'^'^^l
"
"
'
'
V^"
%
'
'
'°^
^j(^V^^j)^'^n'^'^l'---\'^VT"--'^;-l^
n
and therefore the sum in (3.7) is less than
"n'Vr
JljJ^/j'.ij'"'"'"^'"^"-'"-"'"^
'n
We now consider three cases:
(i)
£
a or h = a
(both cannot occur).
(ii)
|£ -
a]
>
q^/2 and
(iii)
]£ -
a|
1
Pp/2 or
h
In
=
j<
-
|h
|h
-
a and i ^ a.
the first case (2.5) applies and
a|
a|
>
q^/2.
^ ^J^
(both cannot occur)
"-^
0,
13
ah
+1
j=^
J
ni
r\
s,-fi=j
^
n
n
n
^^^
j
^.^
or
2,=a
h=a
which goes to
with
n
by Lemma 2.
For the second case we use (2.4) so that
9Q.
<_
dr.
hi
,c ,x
jJ 4.-(c
U
O
II
II
;
z'(h,^,oi))dx^
Qt
where i'(h,Ji,a) contains some primed elements.
tional distribution of x
by c„.
-^
n
Now we comnute the condi-
given the first two variates, represented here
Thus
oo
/
^(h,.,a))dx^
*3(^n'^n'\'
=
*2(^n'S'^h.^(^
"
with (suppressing the primes on the elements of z')
^n^^ha ' ^a)/(l ' ^h£)
^^
-
'L
-
'L
-
4
'
2rh,r^^/,J/(l
By assumption
(3.8)
max (|r^J,
\rj.
\r^,\)<_^^
and using this fact we obtain
4^
*(^^n
"
%^^%^^
14
'<'n-''n'/°ni\(l
provided
is
n
-
36„)/(l + 26„)
taken so large that
1
-
36
Summarizing we have
> 0.
n-1
n
^n
h«,'
>%/2
h-a
n^[l
<
Mn
J
where Yp
C
=
36^)/(1 + 26^).
"
Applying Lemma
2
comoletes the oroof.
The third case is similar to the second one exceot (3.8) is no longer
satisfied.
lrl<6
orir.|<6
a£'
n
'ah'
n
But either
and, of course,
'
kj^
<
6
Conditioning as before we obtain
.
I
n-1
n
I
k.|
I
1
^V^^h.
^-h=j
>i=q„+i
ct=i
<qn/2
h-ct
n-1
.1+3
in-«[l-.(b„v„)]»2(c„.c„.6^W^^|rj
where y^
=
(i
-
26^
6)/(l + 26
-
fi).
Lemma 2 shows that
n-1
,c
n<|)„(c
^
that
n
n
n'^Cl
,6
-
n
)
I
.^^^^^
|r.|
^
and since
2
B
<
(1-6) /2(l+6), Lemma
J
$(b^Yp)] ^ 0.
This completes the proof of Lemma
3.
1
imolies
15
Proof of Theorem
y
<
X.
P{M
1
When y
.
^
Berman's result applies so we consider
x,
Then
<ay+b}
nn-n-^
n—<ax+b,S
n
n
£C^}
P{M^
=
(3.9)
n
+
c^'
(P{Xq, >
I
a=l
P{X^
-
^i
ic
d^; X.
>
.
ic^,
li in,
1
1
1
i
^ a}
^
in,
/
i
a}),
The first term in (3.9) converges to G(y) by Berman's resiult.
in the sum of
(3.9) is of the form treated in Lemma 3.
Each term
Hence we need only
find the limit of
n
(3.10)
P
I
,a,r,,...,r
,c
(c
"
a=l
"
"
'
,0,...,0)
^n
-
P
n
(c
,a,r,
,d
n
n
,.
.
.
,0,...,0)
,r
I
This can be accomplished by using the proof developed in [6] for
q^
a
strong-
mixing sequence.
For each n we are essentially considering a Gaussian
sequence which is
q
n^"^
-
n
-dependent.
1-26
,
Pn =
-nrzF
(a)'
k
(b)
n/k^p^^l,
^
'
If
= "
then
->
n
«>,
'
p
«>
->•
n
n
=
k^(p^.aj
16
and we may apply the block techniques of Theorem
interpret a(.) as
is
function with a(n)
a
used in place of strong-mixing.
1
of [6] provided we
if n >_q
=
The q -deoendence
.
The only work that remains is to
verify that
Pn-T
(3.11)
limk^
(P, - J)P{X^
_I^
>S'Xj.i >c,>
=
0.
When (1.2) holds the proof is exactly the same as the one used for Theorem
3
of [6].
If (1.3) holds then the proof of Theorem 3 of [6] again applies
except that
a
new argument is needed to show that
Pn
(3.12)
n
log n
-2/(l + |r.|)
|r.|
I
n
^
^
0.
By the Cauchv-Schwarz inequality, the square of (3.12) is dominated by
2,
3-4/(U6([p*]))
Pn
.
(log n)^(p^/n)n
"
I
2
r^
J=[o^]^l
which clearly tends to
0.
This completes the proof of Theorem 1.
17
4.
Concluding Remarks
are also valid.
of (Mr
4.T
Lntj
-
)/a
b
n
that given above.
.
The weak convergence results of Theorem 2 of [6]
The convergence of the finite dimensional distributions
n
and (Sr^.n
L^tJ
-
b_^)/a
can be
n
n
proved in a manner -similar to
•
Even if just the one-dimensional
process Mr
.-.
is con-
sidered, it is necessary to verify the convergence of the second maximum
since this is an essential part of the tightness proof given in Theorem
of [6].
We are able to use that tightness proof in this case because it
depends on the form of the limit law for
property.
S
and not on the strong-mixing
2
18
References
[1]
Abramowitz, M. and Stegun, I.
Handbook of Mathematical
(1964).
Functions
National Bureau of Standards Applied Mathematics
Series No. 55.
.
[2]
Limit theorems for the maximum term in
Berman, S. M.
(1964).
502-516.
stationary sequences. Ann. Math. Statist
35
.
Mathematical Methods of Statistics
[3]
Cramer, H.
(1946).
Univ. Press.
[4]
Pickands, J.
Maxima of stationary Gaussian processes.
(1967).
Verw. Qe biete
Z.
Wahrscheinlichkeitstheorie
— —<7 190-223.
—
——
^—^^—^-^^—
—
[5]
[6]
—^——
Slepian,
noise.
D.
.
Princeton
— ^— —
———
——
The one-sided barrier problem for Gaussian
System Tech. J
463-501.
41^
(1962).
Bell
.
A weak convergence theorem for order
Welsch, R. E.
(1970).
statistics from strong-mixing processes. Technical Report No. 57,
Operations Research Center, M.I.T., December, 1970.
To appear
in the Ann. Math. Statist , October, 1971.
.
[7]
Welsch, R. E.
Limit laws for extreme order statistics
(1971).
from strong-mixing processes.
Sloan School of Management Working
Paper 533-71, M.I.T. May, 1971
To appear in the Ann. Math
Statist.
,
.
.
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