HE INSTI
F STATIS
THE CONSOLIDATED UNI
OF NORTH CAROLINA
SINGULARITIES IN THE DISTRIBUTION OF THE
INCREMffi,,rrS OF A sr:'{)OTH PU'NcrION
Donald Gernan
Department of Mathematics and Statistics
University of Massachusetts at Aroberst
and
Department of Statistics
University of North Carolirla at Chapel Hill
"
INSTITUTE OF STATISTICS MIMEO SERIES #1109
l1arch 1977
DEPARTMENT OF STATISTICS
Chapel Hill, North Carolina
SINGULARITIES IN THE DISTRIBUTION OF THE
INCREMENTS OF A SMOOTH FUNCTION
Donald Geman
Department of Mathematics and Statistics
University of Massachusetts at Amherst
and
Department of Statistics
University of North Carolina at Chapel Hill
§l, By the "distribution of the increments" of a Borel function r: 10,11
-+
IR,
~
I mean the measure
1 1
A(B) =
B a Borel set in
m.
f f
o
IB(F(s)-F(t))dsdt
0
A is the convolution of the "occupation measure" )J (B)
m{F-I(B)} with )J(-B); here m in Lebesgue measure.
When)J« m, write o~(x)
for the Radon-Nikodym derivative ~~ (x) (the "local time" of F at x).
)J « m implies A «
Of course
m and
<Xl
(1)
A(x) :: dA (x) =
dm
f
a(y)a(x+y)dy
_<Xl
Although this paper treats only smooth F's (at least·
consists of two general results from [3].
e l ),
the relevant background
Throughout, t}J will denote a nonnegative,
This work was partially supported by National Science Foundation grant
MCS 76-06599 and by the Office of Naval Research Contract NOOOI4-75-C:-0809.
2
Borel measurable function.
I(~;F) =
Then (a) if
4It
Define
1 1
J J ~(F(s)-F(t))dsdt
o
= J~dA ~ 00
0
is even, decreasing on (0,00), and nonintegrab1e on (0,1), then
I(~;F) =oofor any F; (b) 11« m with aEL 2 if and only if I(~;F) < V~c:Ll.
~
00
Now for F differentiable a.e., 11 «
has Lebesgue measure O.
m if and only if DO
==
h: FI (t) = O}
Suppose FECI (i.e. has a continuous derivative, with
the usual conventions about the endpoints) and DO
t- 0 ,
m(D ) = O. Then, as
O
(The additional assumptions made on F
the Theorem states, lim A(x) = 00.
x+O
helow are not needed for this.)
Hence
I(~;F)
= 00 for some
~EL
1
(and so
a~L2) because A « m implies
I(~;F) =
(2)
So, the question arises:
is
J A(x)~(x)dx
for which
~'s
.
- in particular, which monotone ones -
This depends on the nature of the singular points of A.
2
Assume now DO :f 0, F is C
and that F" (t) t- 0 for all tE!)O. Then DO is
I(~;F)
<
oo?
{ai}~=l'
finite, say DO =
0
~ al
~
< a 2<···< aN
{Bi}~=l denote the (distinct) elements of {Ai-A j
1.
Let Ai = F(a i ) and let
} for which there exist
t 1 ,t 2ED O with F"(t l )F"(t 2 ) > 0 and F(t l )-F(t 2 ) = Ai-A j .
about 0 and contains O.
THEOREM.
(3a)
For the version of A(x) given by (1):
A(x) is continuous
O
<
on lR \
0
and
0
1
(3b)
{Bi}~
11m
A(x)
< 11 m
A(x)
-loglx-B·I-log!x-B.
x+B.
1
x+B.
1
Consequently, for
I(~;F)
In particular, if
~
r < 00
~
1
~EL
{B i } is symmetric
1
1
< 00 <=> ~EL {
L
L !loglx-B. I Idx}
i=l
1
.
is even and decreasing on (0,00), then
L.
3
I(~;F) <
§2.
~
1
00
<=>
J ~(x)log
o
l/x dx <
00
•
The fact that the singularities of A occur among the points {A.-A.}
fairly obvious.
1
J
]s
Indeed for Fas above ([4])
(4)
a(x):;:
SEF
-IL
IFI(s)l-
l
({x})
(Since F (DO) has measure zero, i t doesn't matter how a is defined there.)
Clearly a is well-behaved off {A.}, and, in turn, A off {A.-A.}.
1
1
J
(Actually,
(4) is valid for any F such that F' exists a.e., although "sEF-l({x})" must
be replaced by "sEF-l({x}) n {F' exists, finite}" and neither F({IF'I :;: oo})
nOr F({F' doesn't exist}) need have measure 0.)
The Co-Area Theorem [2], applied to the Lipschitz function s,t -)- F(s)-F(t),
leads to this expression for A:
A(x) :;:
(5)
JJ U
[(FI(S))2
+
(F'(t))2]-1/2 H(dsdt)
x
here Ux :;: ((s,t): F(s)-F(t) :;: x} and H is one-dimensional Hausdorff measure in
m2 .. This shows clearly where A might explode. Nonetheless, I will not refer
again to (5), but instead work with the version of A given by (1) with a as
in (4).
That the singularities of A are logarithmic is perhaps not as evident, and
emerged in a curious way.
To get an idea of when
I(~;F)
is finite,
~EL
1
choose a convenient random function X(t,w), 0 :::; t :::; 1, Wd6, with smooth
trajectories and compute the expected value
W -)- I(~;X(',w))~
E(X(s)-X(t)) 2 .
E{I(~;X("w))
of the random variable
For instance, let X(t,w) be Gaussian, mean 0, a 2 (s,t)
Then
4
1 1 00
E{I(ljJ;XCo,W))}
=
.
f J J ljJ(x)[2~a(s,t)]-
2
1
o 0 _00
x
2
} dxdsdt .
2a (s,t)
exp{-
For simplicity, and to insure the differentiability of the sample functions,
suppose there are constants 0 < C < C2 < 00
l
Ys,t.
X(t,w) is stationary, ret) - EXtX O f reO), tf 0, and
(For example,
-r"(O) < 00.)
clls-tl < a(5,t) < c2ls-tl
3
A straightforward computation yields (for ljJ even):
00
J e- y
E{I(ljJ;X(o,w))} < 00 <=>
.
equivalently, ~'I(X)
'I'
f
Y
ya
ljJ(x)dxdy < 00
x
If ljJ is decreasing on (0,00),
'I'
-
o
=lx J0 ljJ(u)du
~"(X) =
2 1
is integrable around the origin, say over [0,1].
th~n
MljJ(X) is the usual maximal function:
1
v
sup
--- J ljJ(y)dy
O<u<x<v<l v-u u
a
< x < 1
1
hence MljJEL [0,1] if and only if ljJELlogL, i.e.
1
J ljJ(x)log+ljJ(x)dx
<00
a
Whether or not ljJ is monotone, Fubini's theorem shows
1
=
Ja ljJ(x) log
1
- dx
x
Consequently, I(ljJ;X(o,w)) < 00 a.s. for any a ~ ljJEL l with
1
1
ljJ(x)log -d, [0,1],
x
and likewise for any stochastic process which satisfies several mild conditions
concerning the distribution of its derivative X'(s,w).
version" of the real-variable theorem above:
A at
a
This is a "stochastic
only the "fixed"singularity of
is picked up; the others - at {B.}\O
- depend on the specific function
1
and w.ill generally occur at any fixed point Xo with probability
o.
Rounding out the picture, it follows from a theorem of Bul inskaya
[1] that the hypotheses of the theorem are valid for almost every sample function~
of a stochastic process
X(t,w) for which: (i)
2
X(o,w) is C
a
.s.,
5
tit
(ii)
for each 0
t and
x.
~
t
5
1, XI(t,W) has a density Pt(x) which is bounded in
Condition (ii) guarantees that {t: XI (t ,w) = X" (t ,w) = O} is empty a. s.
earlier statement "I(l/J;X(o,w)) <
ened to "I(l/J;X(o,w)) <
00
Our
a.s. for any l/J+ in LlogL" can then he strength-
00
for all1jJ+ in LlogL, a.s.," Le. the exceptional w-set no
longer depends on the particular l/J.
Here is the proof of the theorem, which uses little else than ordinary
§3.
IV'I/\
calculus.
Recall that A is the version of dA/dm given by
00
A(x)
=
J
_00
(i)
I . . . /FI(S)
sEF-I({x+y})
A is continuous off {Bi}i.
,-I
SEF
_II
IFI(s)/-ldy,
({y})
_00
<x <
00.
I will show that A is continuous on
N
L
{A.-A.}. . l\{B'}I; the proof of continuity at xEffi.\{A.-A.} goes about the
1
J I,J=
1
1
J
same, except is easier.
Let AO = inf F(s), ~+l= s~p F(s),and vex) = Card{s:F(s)=x}
s
Card{D
o}
-1
and F
small
£
<
00.
~ 1 +
First, notice that a is continuous off {Ai}~+l because FI
(defined piecewise) are continuous, and because v(x+£) = vex) for all
if X~{Ai}6+1.
Now fix Ai-Aj~Bk}i, 1 ~i,j ~ N, and let (k~,r~), ~ = l, ... ,q, be those
pairs of integers among {I,2, ... ,N} for which A -A
= A. -A..
Assume
k~ r~
1
J
F"(a i ) < 0 < F" (a j ); then F" (a k) < 0 (resp. F" (a k ) > 0) for each 1 ~ k ~ N
with Ak = A. (resp. A = A.). It follows that a(A. -)= lim a(A.-E) and
k
J
1
J
£+0
J
a(A.+)= lim a(A.+c) exist, finite. The same argument applies to each A ,A
k~ r~
1
£+0
1
and yields:
(*)
a(A +) <
k
00
,
a(A -) <
1
00,
r~
~
~ ~ ~
q.
N
L
Since A(x) is an even function and {A.-A.} . . 1\{B'}1 is symmetric about
1
J I,J=·
1
6
0, it will be enough to check that A is right-continuous at A.-A ..
1
K(x,y)
=
J
Set
a(y)a(y+x+A.-A.) and let
1
J
N
To = U (Ak-o,Ak+o),
k=l
also, let n > 0 be the distance from A.-A. to {A -A }, (n,m) ~ (ko,r o).
1
J
n m
Yv
Yv
Then
A (x+A. -A.)
1
J
where the Ar '
Q,
=
Q,Er~{I,
... ,q}, are distinct and 0 is small enough that the
intervals (A
-o,A +0), Q,Ef, are disjoint.
rQ,
rQ,
If 0 ~ x < 0/2 and 0 < n/2, YEw~nTo ~> y+Ai-AjET~ => y+X+Ai-AjET~/2·In
particular,
sup a(y+x+A.-A.) < 00 for such XIS. Consequently, recalling
c
1
J
YEWO nT 0
I
that uEL and a is continuous a.e., PI (x) + PI(O)< 00 as x~O (dominated
convergence theorem).
Finally,
Similarly, P2 (x) <,00 'Vx ~ 0 and
00
~ sup a(y)! \a(x+y)-a(y) \dy + 0 as x~O.
c
yET 0
00
A
'\ +0
rQ,
JQ, a(y-A.+A.)a(y+x)dy
= !K(x,y)dy +
A
1
J
A -0
kQ,
rQ,
which converges to P Q,(O) < 00 as
3•
x~O
by using (*) and arguing as above with
7
e
Next, F l oF- 1 satisfies upper and lower Holder conditions of order 1/2 at each
A., 1
:0::
i :s; N.
For convenience, assume 0 < a O < aN < 1; the other cases only
1.
For each aiED O and sE[a,l] there are numhers
need some additional notation.
-
E;"E;,
s
s
between sand a.l
with FI (s)
:=
1
It follows that there are constants 0 < Cl ,C 2 ,C 3 'C 4 <
for each 1 :s; i :s; Nand 0 :s;
00
I
and a
60>
0 such that
sE(a.-o,a.+o)
c2Is-a./
:s; IFI (s)
1
(6b)
2
2
c /s-a.1
1 :s; IF(s)-A. I :s; c3Is-a. 1 ,
1 1
4
_Cl/y_A.ll/2<_IFA . ()
y -a.
1
:s;. clls-a.1
'
1
= [a.,a.
1]'
1
1+
A
Let F.1 denote the inverse of F on J.1
3
2
60 ,
(6a)
(7)
-
FII(E;,S) (s-a ·) andF(s)
- A. := -2F"('; ) (s-a.)
S
1
1
I
1
1
1
1
s€(a.-o,a.+o) .
1
1 :s; i ~ N-l.
l 2
<-1.I_A·l
- C Y 1· / ,
4
1
1
From (6b)
yE(A.-o,A.+o)nF(J.)
I I I
A
11
,1/2
I
C y-A i +l
:s; Fi(y) -a i +1 I :s; .-!-Iy-A
i+1 ,1/2 ' yE(A.1+ l-o,A.]~~.1+8)nF(J 1·.).
C
4
3
Let Dei,o)
:=
(A.,A.+o)
if F"Ca.)
> 0,
11·
1
=
Combining (6a) and (7), and reducing 0
a <C ' C <
S 6
00
0
(A.-o,A.)
i f F"(a.) < 0,1 s i:s; N.
I I I
if necessary, there are constants
such that for each 1 s i s N-I, 0 s 00 ,
Csly-A.
1 2
IFI(~.
1 /
s
(y)) 1 s C6 /y-A. 1
I I I
(8)
and likewise (in case aN
= 1)
1 2
/ ,
YED(i,o)
with A.,
D(i,o) replaced by A. l' D(i+l,o).
1 1 +
We can assume that for each i,j and each small 0, either D(i,a)
D(i,o)nD(j,o)
o'
inverses F
=~.
FN,
Defining J
O
= [O,a ], I
I
N
=
[~,l]
= D(j,o)
or
and the corresponding
it is clear that (8) extends to FloFo and FloFN at the
appropriate places. (By the way, both inequalities in (8) depend on F" ". 0 on
8
(ii)
lim J\.(x)/-loglx-B.I > 0, 1
1
x+B .
$
i
L.
$
Suppose B. = An-A k , 1:; Q"
1
k
$
N,
)t"
1
e
and F"(a ) < 0, F"(aQ,) < 0; the other case, namely F"(aQ,), FIf(a ) > 0 is the same ..
k
k
00
f
J\.(x+B i ) =
_00
a(y+x+AQ,)a(y+Ak)dy
~
J
-Ixl
-E
a (y+x+AQ,)a (y+A k ) dy,
Now for E small, the conditions Ixl <
Y+X+AQ,ED(Q"oO) and y+AkED(k,oO)'
2
J\.(x+B i ) >~ C5
£
Ixl <
E
•
and -E < Y < -Ixl together imply that
Consequently,
J- Ixl ,y+x ,-1/2 1y ,-1/2 dy
-E
2
2/E2_EX + 2E-X
= C5 log - - - - - - - 2
2/ x -lxlx + 2lxl-x
~
1
TXT '
C log
for all small x, for some C > O.
(iii)
J\.(x)
$
canst. x[1 + I~lloglx-Bill]\'x. (This is equivalent to the "lim"
part of (3a).) Evidently,
a(y)
YET~,
Let
Off To, a is bounded.
u
say A.-o
< y < A.+o, yEF[O.l].
1
1
Keeping (8)
in mind and that non-identical D(j,O)1 S are disjoint:
a(y)
=.
I
IF(J.)(y)IF1oF.(y)
J:A.=A.
J
J
J
1
$
$
,-I
+ .
I
J : A.I t-A.J
IF(J.)(y)rF'o~j(Y) 1-1
J
.
.
N
V(y)suPIF'(s) 1-1, H8 = F-1 l n (A.-O,A.+o) c J
i=l 1
1
sEH o
N
canst. x [1 + I ly_A.I- 1/ 2 ]
V(y) C IY-A ,-1/2
s
i
. 1
1=
+
1
9
N
Let V
= F[O,I]
A(x) =
and
U = U V-A. , which is bounded.
.' I
1=
1
J a(y)a(x+y)dy
V
N
J
/y-A.
::; const. x [l + 2 I
'IV
1
1=
N
::; const.x[l +
J
I
, J=
, I U
1,
N
I- I / 2dy
+
I.
J Iy-A. I-I/21y+x~A, ,-1/2 dy ]
. 'IV
1,J=
J.
1
ly+A.-A, ,-1/2Iy+x/- 1 / 2dy]
J 1
N
::; const.x[l +
.
I
Iloglx-(A,-A·)I I]
. I
1, J
J
1
=
I
o (log 1sT)
since
As for (3b), let H(x) = 1 +
as
E: -+
I~=llloglx-Bili
O.
Then I (ljJ; F) <
.
00
1
VqJEL (lIdm)
if and only if
00
J
_00
~~~~
if and only if essxsup
this is the same as
ljJ(x)
H(x) A(x)dx <
<
s~p ~~~~
equivalent to "I(ljJ;F) <
00
VljJEL (dx) ,
Since A, H are continuous from
00.
<
1
00
00
m. to m l{OO},
In other words, the "lim" part of (3a) is
VljJdl(Hdm)."
Now if I(ljJ;F) <
00
and ljJEL1(dx), then
it is easy to see, using the "lim" part of (3a) that ljJH is integrable.
The
last statement of the theorem follows from (3 b) and the aforementioned fact
that I(ljJ;F) <
~
00
and ljJ+ imply ljJELl[O,l].
2
Let F(t) = t .
Then DO = {B } = {oJ and
i
A(x) =
1:..
2
log{ 1 +
1rJTI}
/fXT
Ix I :,;
1 .
For F(t) = sin 27ft, A(x) is an elliptic integral (of the first kind).
I would give
more examples, especially in "closed form" and with L > 1, if I could; the computat ions (even for F a third degree polynomial) are formidable.
10
References
1.
E.V. Bu1inskaya, On the mean number of crossings of a level by a stationary
Gaussian process, Teoriya Veroyatnostei i ee Primeneniya, 6 (1961),
pp. 474-477.
2.
H. Federer, Geometric Measure Theory, Berlin: Springer-Verlag (1969).
3.
D. Geman and J. Horowitz, Local times for real and random functions, Duke
Math. J., 43 (1976), pp. 809-828.
4.
D. Geman and J. Horowitz, Occupation-times for functions with countable level
sets and the regeneration of stationary processes, Z. Wahrschcinlichkeltstheorie verw. Geb. 35 (1976), pp. 189-211.
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I. REPORT NUMBER
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GOVT ACCESSION NO. 3.
TITLE (and Subtitle)
Singularities in the Distribution of the
Increments of a Smooth Function
RECIPIENT'S CATALOG NUMBER
S.
TYPE OF REPORT & PERIOD COVERED
6.
PERFORMING ORG. REPORT NUMBER
Technical
Mimeo Series 111109
7.
AUTHOR(s)
B.
Donald Geman
9.
CONTRACT OR GRANT NUMBER(s)
ONR NOOOI4-7S-C-0809
PERFORMING ORGANIZATION NAME AND ADDRESS
10.
PROGRAM ELEMENT. PROJECT. 'tASK
AREA & WORK UNIT NUMBERS
12.
REPORT DATE
Department of Statistics
University of North Carolina
Chanel Hill NC 27 i; 1Ll.
II.
CONTROLLING OFFICE NAME AND ADDRESS
March 1977
13. NUMBER OF PAGES
10
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SUPPl.EMENTARY NOTES
19.
K
20.
Y WORDS (Continue on reverse eide If necessary and Identify by block number)
2
C function,
local time, distribution of the increments
ABSTRACT (Continue on reverse side If nece .... ary and identify by block number).
Let F(t), O~t~l, be a real function with two continuous derivatives such that
{F' =F"=O} is empty. Then B + meas.{(s,t):F(s)-F(t)EB} in absolutely continuous;
its density is continuous on IR\{B.}, {B.} - {y: y=F(t )-F(t ), F'(t ) =
1
1
l
2
1
F' (t 2) = 0, Filet F"(t > a}, and has a logarithmic singularity at each B.•
1
2
1
DO
FORM
1 JAN 73
1473
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