A R K I V F O R M A T E M A T I K B a n d 6 n r 20
1.65 11 1 R e a d 8 S e p t e m b e r 1965
On the intersections between the trajectories o f a n o r m a l stationary stochastic process and a high level
By HARALD CRkMER
l . Introduction
L e t ~(t) be a real-valued, normal and stationary stochastic process with a continuous time p a r a m e t e r t, v a r y i n g over ( - ~ , co). Suppose t h a t E ~ ( t ) = 0 for all t, a n d t h a t the covariance function
r(t -- u) = E~(t) ~(u) satisfies the following two conditions:
as t--> 0, a n d
1_
]r(t) = 1 - 2! ~2t2 + ~I. ~4t4 § ~
r(t) = o(I t l -~)
(1)
(2)
for some a > 0 , as t - + _ c o . We m a y , of course, assume ~ < 1 .
I t follows 1 from (1) t h a t there exists an equivalent version of $(t) having, with probability one, a continuous sample function derivative ~'(t), a n d it will be supposed t h a t ~(t) has, if required, been replaced b y this equivalent version.
L e t u > 0 be given, a n d consider the intersections between the trajectories
= ~(t) of the ~ process a n d the horizontal line, or "level", B = u during some finite time interval, say 0 < t < T. I t follows f r o m a theorem due to Bulinskaja [2] that, with p r o b a b i l i t y one, there are only a finite n u m b e r of such inter- sections, and also that, with p r o b a b i l i t y one, there is no point of t a n g e n c y between the trajectory B = ~(t) and the level U = u during the interval (0, T).
Thus, with p r o b a b i l i t y one, every point with ~ ( t ) = u can be classified as an
"upcrossing" or a "downcrossing" of the level u, according as ~'(t) is positive or negative.
The present p a p e r will be concerned with the upcrossings, a n d their asymp- totic distribution in time, as the level u becomes large. I t will be obvious t h a t the case of a large negative u, as well as the corresponding problem for the downcrossings, can be t r e a t e d in the same way.
The uperossings m a y be regarded as a stationary stream o~ random events (cs
1 W i t h respect to the general t h e o r y of t h e n o r m a l s t a t i o n a r y process a n d its s a m p l e f u n c t i o n s we refer to t h e f o r t h c o m i n g book [4] b y Cram~r a n d L e a d b e t t e r .
H. CRAM~R, Stationary stochastic process and high level
e . g . Khintchine's book [5]), a n d it is well known t h a t the simplest case of such a s t r e a m occurs when the successive events f o r m a Poisson process.
However, a necessary condition for this case is (Khintchine, I.c., pp. 11-12) t h a t the numbers of events occurring in a n y two disjoint time intervals should be independent r a n d o m variables, a n d it is readily seen t h a t this condition can- n o t be expected to be satisfied b y the stream of upcrossings. On the other hand, it seems intuitively plausible t h a t the independence condition should be a t least a p p r o x i m a t e l y satisfied when the level u becomes v e r y large, provided t h a t values of ~(t) lying far a p a r t on the time scale can be supposed to be only weakly dependent.
Accordingly it m a y be supposed that, subject to appropriate conditions on
~(t), the stream of upcrossings will t e n d to form a Poisson process as the level u tends to infinity.
T h a t this is actually so was first proved b y Volkonskij a n d R o z a n o v in their remarkable joint p a p e r [8]. T h e y assumed, in addition to the condition (1) above, t h a t ~(t) satisfies the so-called condition o/strong mixing. This is a fairly restrictive condition, as can be seen e.g. from the analysis of the strong mixing concept given b y Kolmogorov a n d R o z a n o v [6]. Moreover, in an actual case it will n o t always be easy to decide whether :the condition is satisfied or not. On the other hand, various interesting properties of the ~(t) trajectories follow as corollaries from the a s y m p t o t i c Poisson character of the stream of upcrossings (ef. Cram6r [3], Cram6r a n d L e a d b e t t e r [4]), so t h a t it seems highly desirable to prove the latter p r o p e r t y under simpler a n d less restrictive conditions.
I t will be shown in the present p a p e r t h a t it is possible to replace the con- dition of strong mixing b y the condition (2) above. This is considerably more general, a n d also simpler to deal with in m o s t applications.
I a m indebted to Dr. Yu. K. Belajev for stimulating conversations a b o u t the problem t r e a t e d in this paper.
2. T h e m a i n t h e o r e m
The n u m b e r of upcrossings of ~(t) with the level u during the time interval (s, t) will be denoted b y N(s,t). This is a r a n d o m variable which, b y the above remarks, is finite with p r o b a b i l i t y one. W h e n s = 0, we write simply N(t) instead of N(0,t). F r o m the stationarity and the condition (1) it follows (Bulinskaja [2], Cram6r and Leadbetter [4], Ch. 10) t h a t the m e a n value of N(s,t) is, for 8 < t ,
E N ( 8 , t) = E N ( t - 8) = / , ( t - s), (3)
where # = EN(1) = ( ~ e- u,12. (4)
zT~
The q u a n t i t y /~ will p l a y an i m p o r t a n t p a r t in the sequel. We note t h a t /~ is a function of the level u, a n d t h a t /t tends to zero as u tends to infinity. F r o m (3) we obtain for a n y T > 0
EN(~/~) = T.
ARKiV FOR MATEMATIK. B d 6 nr 20 For the s t u d y of the stream of upcrossings, it will then seem n a t u r a l to choose 1//~ as a scaling unit of time, thus replacing t b y T//~. We might expect, e.g., t h a t the probability distribution of fr will tend to some limiting form when /x tends to zero while ~ remains fixed. This is in f a c t the case, as shown b y the following theorem, first p r o v e d b y Volkonskij a n d R o z a n o v under more restrictive conditions, as mentioned above.
Theorem. Suppose that the normal and stationary process ~(t) satis/ies the conditions (1) and (2). Let (%,b~), ..., (aj, bj) be disjoint time intervals depending on u in such a way that,/or i = 1 . . . ],
b~ - a~ = vJlu ,
the integer j and the positive numbers T1,... ~ being independent o / u . Let lc 1 . . . kj be non--negative integers independent o / u . Then
] T k ~
lim P { N(a~, b~) = k~ for i = 1 . . . ?" } = ] 7 --~ e--7;' @
Thus, when time is measured in units of 1/~t, the stream of upcrossings will a s y m p t o t i c a l l y behave as a Poisson process as the level u tends to infinity.
We shall first give the proof for the case i = 1, when there is one single interval (a,b) of length b - a - = ~ / / x . Owing to the s t a t i o n a r i t y it is sufficient to consider the interval (O,v/#). Writing
T = ~ J ~ t , P k = P { N ( T ) - = k } , (5) we then h a v e to prove the relation
T k
_ -7;
lim Pk -- ~ e (6)
u - - ~ o o
for a n y given T > 0 a n d non-negative integer k, b o t h independent of u. Once this has been achieved, the proof of the general case will follow in a com- p a r a t i v e l y simple way.
The proof of (6) is r a t h e r long, a n d will be broken up in a series of lemmas.
I n the following section we shall introduce some notations t h a t will be used in the course of the proof. The l e m m a s will be given in sections 4 a n d 5, while section 6 contains the proof of the case ?'= 1 of the theorem, a n d section 7 the proof of the general case.
3. N o t a t i o n s
The level u will be regarded as a variable tending to infinity, a n d we m u s t now introduce various functions of u. I t will be practical to define t h e m as functions of /~, where /~ is the function of u given b y (4). Writing as usual [x] for the greatest integer ~< x, we define
M = [T/#a]. n = [ M / ( m 1 + m2)] + 1. (7)
~. Cl~Mf~a, Stationary stochastic process and high level Here fl is a n y n u m b e r satisfying the relation
0 < ( ~ + 4 ) f l < ~ < l , (s)
where a is the constant occurring in the condition (2), while /c is the integer occurring in (6). W e further write
q = T / M , t 1 = m l q , 12 = m2q , (9)
a n d divide the interval (0,T) on the time axis into subintervals, alternatively of length t I a n d t2, starting from the origin. We shall refer to these subinter- vals as t I- a n d t~-intervals respectively, the former being regarded as closed a n d the latter as open. E a c h t~-interval (i = 1,2) consists of m~ subintervals of length q. The whole interval (0,T), which consists of M intervals of length q, is covered b y n pairs of t 1- a n d t~-intervals, the n t h pair being possibly incomplete. A n y two distinct tl-intervals are separated b y an interval of length a t least equal to t 2. Art i m p o r t a n t use will be made of this r e m a r k in the proof of L e m m a 5 below.
The quantities defined b y (7) a n d (9) are all functions of u. I t will be prac- tical to express their order of magnitude for large u in terms of /~. The fol- lowing relations are easily obtained from (5), (7) a n d (9):
q~#~, n ~ ~ # -~,
(10) t 1 ~ / z ~ -1, t 2 ~/z2~ -1.
f
We now define a stochastic process }q(t) b y taking
~a (v~) = ~(~q)
for all integers v, a n d determining ~q b y linear interpolation in the interval between two consequtive vq. To a n y sample function of the ~(t) process will then correspond a sample function of
~q(t),
which is graphically represented b y the broken line joining the points [vq,~(vq)]. For the n u m b e r of upcrossings of this broken line with the u level we use the notations Hq (s, t) and H a (t), corre- sponding to N(s,t) and H(t). The probability corresponding to Pk as defined b y (5) isP(ka)= p { N q ( T ) = k }. (11)
4. L e m m a s 1 - 3
Throughout the rest of the p a p e r we assume t h a t the ~(t) process satisfies the conditions of the above theorem.
L e m m a 1. I / T and q are given by (5) and (9), ~ and Ic b e i n g / i x e d as be/ore, we have
lim (Pk - P(k q)) = 0.
Evidently N q ( T ) < ~ N ( T ) . W e shall prove t h a t the non-negative a n d integer- valued r a n d o m variable N ( T ) - H q ( T ) converges in first order m e a n to zero,
ARKIV FOR MATEMATIK, B d 6 n r 2 0 as u - + o o . I t then follows t h a t the probability t h a t
N(T)-Nq(T)takes
a n yvalue different from zero will tend to zero, and so the lemma will be proved.
B y (3) and (5), the mean value of the number /V(T) of uperossings of ~(t) in (0,T) is for every u
EN(T)=T#=T.
I t will now be proved t h a t the mean value
ENq(T)
tends to the limit T as u - + oo, so t h a t we havelim E {
N(T) - Nq (T) } = O.
U - > C o
Since N ( T ) - N q ( T ) is non-negative, this implies convergence in first order mean to zero, so t h a t by the above remark the lemma will be proved.
Consider first the number Nq(q) of upcrossings of ~q(t) in the interval (0,q).
This number is one, if ~ ( 0 ) < u < ~(q), and otherwise zero, so t h a t
ENq(q)
= P { ~(0) < u < ~(q)}.N o w ~(0) and $(q) have a joint normal density function, with unit variances a n d correlation coefficient
r=r(q).
B y (1) and (10) we haver(q) = 1 - ~ ~ q ~ + O(q4).
(12)
For the probability t h a t ~(0)< u and ~(q)> u we obtain b y a standard trans- formation
where as usual
ENq(q) = V~ f ~ e ~,12ap[/_~_r2)dx
, / u - r x \O(x)=i/~f~coe t~/2dt.
:By some straightforward evaluation of the integral from u + ( 1 - r)~ to infinity we obtain, using (12), and denoting b y K an unspecified positive constant,
1
E N q ( q ) = / ~ j u [~/~_r~}
+ O[exp ( - .For the first term in the second member we obtain, using again (12), the ex- pression
1 - u ~ / 2
e (1 +~u+(1-r)~(O(u-r.cid x
\V~-rV
V1 - r ~
r~/~ e ~'/2(l+O(q89176 O(y'dy+O(qu')=~22-~:qe-~'/2'l+O(q 89
so that
ENq (q) = q# + o(q#).
H. Cr, AM~R, Stationary stochastic process and high level
I f v denotes an integer, which m a y tend to infinity with u, it t h e n follows from the stationarity t h a t we h a v e
ENq(vq) = vq/~ + o@q/~). (13)
I n particular, taking v = M we obtain from (9) a n d (5) E~V. (T) = T ~ + o ( ~ ) -~ 3.
According to the above remarks, this proves the lemma.
We now consider the n u m b e r /~q(tl) of ~q uperossings in an interval of length t l = m l q , observing t h a t b y (10) we h a v e t l ~ ~-1.
L e m m a 2. We have
lira E { Nq(tl) [Nq(t~) - l]} 0.
B y (3) we have EN(tl)=$1/s while (13) gives for v = m 1 ENq (tl) = t l ~ "~- O(tl/~ ) ~" EN(tl).
Further, since Nq(tl) ~ N(tl) ,
E { Nq (t) [Nq (ti) - 1]} ~< E { N(tl) [N(tl) - 1 }.
The t r u t h of the l e m m a will then follow from the corresponding relation with
~Vq replaced b y N. N o w this latter relation is identical with the relation proved b y Volkonskij and R o z a n o v [8] in their L e m m a 3.4. I t is proved b y t h e m without a n y mixing hypothesis, assuming only t h a t ~(t) is regular (or purely non-deterministic) a n d t h a t r(t) has a fourth order derivative a t t=O. Their proof is valid without a n y modification whatever, if these conditions are replaced b y our conditions (1) and (2). Thus we m a y refer to their paper for the proof of this lemma. We note t h a t the proof is based on the i m p o r t a n t work of S. O. Rice [7].
L e m m a 3. A s u--> ~ , we have
P { N q ( t ~ ) = 0 } = 1 - q + o ( q ) , P { Nq(t~) = l } = q + o(q), P { N q (~1) > 1 } = O(q).
For a n y r a n d o m variable v taking only non-negative integral values we have, writing ni = P { v = i } a n d assuming Ev 2 < ~ ,
E~' = 7~ 1 -{- 27~ 2 -}- 37~ 3 -{- ..., Er(v - l) = 2 ~ 2 + 67~3-}- ....
and consequently
E~) -- E~,('p - l ) ~ 17~ 1 ~ 1 -;Tg 0 ~ E y . (14) T a k i n g ~ ) = ~ q ( t l ) , and observing t h a t b y (10) we have E N q ( t l ) , ~ t l # ~ q , the t r u t h of the lemma follows directly f r o m L e m m a 2.
ARKIV F6R MATEMATIK, B d 6 n r 2 0 5 . L e m m a s 4 - 5
F o r each r = l , 2 , . . . , n , we now define the following events, i.e. the sets of all ~(t) sample functions satisfying the conditions written between the brackets:
Cr= { e x a c t l y one ~q upcrossing in the r t h tl-interval}, dr= { a t least one ~q upcrossing in the r t h tl-interval }, er = { ~(~(1) > u for a t least one vq in the r t h t~-interval }.
Further, let C~ denote the event t h a t e r occurs in exactly k of the tx-intervals in (0,T), while the c o m p l e m e n t a r y event c~* occurs in the n - k others. D k a n d E~ are defined in the corresponding way, using respectively dr and er instead
o f C r.
L e m m a 4. For the probability P(~q) defined by (11) we have lira [Piq)- P { } ] = O .
u - - > o ~
We shall prove t h a t each of the differences P(k q) - P { Ck }, P { Ck } - P { Dk } a n d P { D k } - P { E k } tends to zero as u - - > ~ .
B y (13) a n d (14) the probability of a t least one ~q upcrossing in an interval of length t 2 is at m o s t ENq(t2)-~t21~+o(t2#). Thus the probability of a t least one ~q upcrossing in at least one of the n t2-intervals in (0,T) is b y (10)
O(nt~/u ) = O(lu~),
a n d thus tends to zero as u--> co. I t follows t h a t we h a v e
P(kq)--P{total n u m b e r of Sq upcrossings in all n t l - i n t e r v a l s = k } - + 0 . (15) On the other hand, b y the s t a t i o n a r i t y of ~(t), L e m m a 3 remains true if Nq(tl) is replaced b y the n u m b e r of ~q upcrossings in a n y particular t~-interval. Since the interval (0,T) contains n of these intervals, it follows from (10) t h a t the probability of more t h a n one ~q upcrossing in at least one of the tl-intervals is o(nq)=o(1), and Chus tends to zero as u - - ~ .
F r o m (15) and the last remark, it now readily follows t h a t the differences P(k q ) - p { C k } and P { C k } - - P { D k } both tend to zero as u - - ~ . I t thus only remains to show t h a t this is true also for P { D k } - P { E ~ } .
B y the definitions of the events D~ and Ek we h a v e p { D k } = h p { d r l . . . d ,t* d* "t I
-rk~sl . . . . "-~ J' (16)
P{E i=hP{% }, f
the summations being extended over all ( ; ) groups of/c different subscripts r I . . . . ;r~
selected among the numbers 1 , . . . , n , while in each case s 1 .. . . . s~-k are the remaining n - k subscripts.
Let vrq denote the left endpoint of the rth tl-interval, a n d denote b y gr the event
= { u }
H. CB_~f~R, Stationary stochastic process and high level of probability
Then for every r = 1 , . . . , n
so t h a t
d r c e r a n d e r - d r ~ g r ,
(17)
Similarly e* c d*, a n d
d* - - e r = e r - d r C g r , *
B y a simple recursive a r g u m e n t (16) then yields, using (8) a n d (10), /nk*~ \ = 0 [exp
(
1 - -which proves the lemma.
* E a c h of these is B y definition, Ek is composed of certain events er a n d es.
associated with one particular t~-interval, and it h a s been observed above t h a t a n y two tl-intervals are separated b y an interval which is of length>~ t2, a n d thus tends to infinity with u. B y means of the condition (2) it will now be shown t h a t the component events of Ek are asymptotically independent, as u--> co. Moreover, owing to stationarity, the probability
p = P { e , } (18)
is independent of r, so t h a t b y (16) the asymptotic independence will be ex- pressed b y the following lemma.
L e m m a 5. The probability p being de/ined by (18) we have, as u - ~ ~ ,
I n order to prove this lemma, we consider the points vq on the time axis for all integers v such t h a t rq belongs to one of the tl-intervals in (0, T). E a c h tl-interval , which we regard as closed, contains m 1 § 1 points vq, a n d there are n - 1 complete a n d one possibly incomplete such interval in (0,T). I f L is the t o t a l n u m b e r of points vq in all tl-intervals, we thus have
(n - 1) (m 1 + 1) < L <~ n(m 1 + 1).
L e t ~1 . . . ~L be the r a n d o m variables ~(v~) corresponding to all these L points
A R K I V F O R M A T E M A T I K . B d 6 n r 20
vq, ordered according to increasing v. The ~ sequence will consist of n groups, each corresponding to one particular tl-interval.
Further, let /1
(Yl
. . . . ,YL)
be the L-dimensional normal probability density of U1 .. . . ,~L, a n d let A 1 be the corresponding covariance m a t r i x . (Our reasons for using the subscript 1 here a n d in the sequel will presently appear.) F r o m (16) we obtainP ~ Ek ~= JEt-~/~dy= : ,]eiw..~'~,.../ldY'
(19)where the a b b r e v i a t e d notation should be easily understood, the s u m m a t i o n being extended as explained after (16).
L e t us now consider one particular t e r m of the sum in the last m e m b e r of (19), say the t e r m where the group of subscripts r 1 .. . . . rk coincides with the integers 1 . . . k. I t will be readily seen t h a t a n y other t e r m can be treated in the same w a y as we propose to do with this one. This t e r m is
F(1)
= fo/ldy,
where G denotes the set
G = e 1 . ~ eke~+l... * e n ~
F(1) m a y be regarded as a function of the covariances which are elements of the m a t r i x A 1. L e t us consider in particular the dependence of F(1) on those covariances
Q~j=E~
which correspond to variables ~ a n d ~jbelonging to di/- ]erent tl-intervals.
I f all covariances Qij having this character are replaced b y~j=h~j,
with 0~<h~<l, while all other elements of A 1 remain unchanged, the resulting m a t r i x will beAh = hA 1 + (1 - h) A 0, (20)
while the density function /1 will be replaced b y a certain function/h. E v i d e n t l y /0, corresponding to the covariance m a t r i x A0, will be the normal density func- tion t h a t would a p p l y if the groups of variables ~ belonging to different t 1- intervals were all m u t u a l l y independent, while the joint distribution within each group were the same as before.
Thus A 1 a n d A 0 are b o t h positive definite, a n d it then follows from (20) t h a t the same is true for An, so t h a t / h is always a normal p r o b a b i l i t y density. Writing
F(h) = fG/ dy,
it follows from the r e m a r k s just m a d e t h a t we have
fe fe fe fe
* ' *F(O)= ,/~ ~/~
~+l/~,/~ ...P{en}.
B y s t a t i o n a r i t y this reduces to
_F(0) =p~(1 _ p ) n - k , (21)
where p is given b y (18).
H. Cm~M~R, Stationary stochastic process and high level
We shall now evaluate the difference F ( 1 ) - F ( 0 ) b y a development of a m e t h o d used b y S. M. B e r m a n [1]. We note t h a t for a n y normal density func- tion /(x I . . . xn) with zero means a n d covariances r~r we h a v e
Or~ Ox~Ox~ ~
I n our case, [n(Ya,...,Yr.) is a normal density, depending on h through the covariances 2~ = h ~ . Hence
F'(h)= f d~h dY ~ O/h du ( O2fu
dy, (22)the s u m m a t i o n being extended over all i, ~ such t h a t ~l and ~j belong to dif- ferent tl-intervals. W i t h respect to the integral over the set G = e 1... eke~+l...e*~
occurring in the last sum in (22), we have to distinguish three different cases.
Case A . W h e n ~ a n d ~}j b o t h belong to t~-intervals of subscripts >/c, say to the ts-intervals of subscripts k + 1 a n d /c + 2 respectively, integration with respect to y~ a n d yj has to be performed over e~+l a n d e*+2 respectively. B y definition of the sets er, both y~ and yj thus have to be integrated over ( - c ~ , 0 ) , a n d so we obtain b y direct integration with respect to yi a n d yj
f
a ~2/h d y -- f h(y,=yj=u)dy'.
(23)The notation used in the last integral is to be understood so t h a t we have to t a k e y~ = y j = u in [h, a n d then integrate with respect to all y's expect y~ and yj. As /h > 0 always, we have
J ~ ay~ 0yj ... fh (yi = yj = u) dy .
The last integral, where all the y's except y~ a n d yj are integrated out, yields the joint density function of the r a n d o m variables corresponding to ~ a n d ~j in the normal distribution with covariance m a t r i x Ah, for the values y~ = yj = u, so t h a t
O< ~e[h d y < exp
[-u~/(l§
(24)~yi ~yj 2~(1 - h 2 ff~)89
Case B. L e t now ~/i a n d ~/j b o t h belong to tl-intervals of subscripts ~/c, say to those of subscripts 1 a n d 2 respectively. Then integration with respect to each of the groups of variables to which Yi a n d yj belong has to be performed over e I a n d e~ respectively. B y definition, el is the set of all points in the y space such t h a t at least one of the y's associated with the first tl-interval exceeds u, a n d correspondingly for %. Some reflection will then show t h a t the integration indicated in the first m e m b e r of (23) can still be carried out directly, and yields the same result, with the only difference t h a t in the second m e m b e r
ARKIV FOR MATEMATIK. Bd 6 nr 20 of (23) the integration has to be performed over a set
G',
obtained from G b y replacing e 1 a n d e 2 b y e~ a n d e*2 respectively. I t follows t h a t the inequality (24) still holds.Case C.
Finally we have the case when ~ a n d ~/s belong to tl-intervals of different kinds, say to the first a n d the ( k + l ) s t respectively. As before the integration in the first m e m b e r of (23) can be carried out directly. I n this case, however, we obtain ~ the relation (23) with a changed sign of the second m e m b e r , a n d e 1 replaced b y et in the expression of the domain of integration. I n this case we thus obtain the inequality (24) with changed inequality signs.Thus in all three cases we h a v e the inequality
fa ~2/h dy < 1
e,J exp (25)
N o w ~j is the covariance between the variables ~,=}(~,q) a n d ~j=}(vjq), where the points ~ q and vjq belong to different tl-intervals, a n d are thus sep- a r a t e d b y an interval of length a t least equal to t 2. B y the condition (2) we t h e n h a v e
l e,J I = I r ( ~ , q - r~q)l <
Kt2 ~,
where as usual K denotes an unspecified positive constant. Further, there are less t h a n
L2<<.n2(ml+l) ~
covariances ~ts. Owing to s t a t i o n a r i t y some of the p~j are equal, b u t it is easily seen t h a t this does not affect our argument. I t then follows f r o m (22) a n d (25), using (7) a n d (10), t h a t we h a v e[F,(h)l < . . 2 ~t-= - = , < /~xn ml 2 e K # ~-4fl,
This holds for a n y of the ( ; ) t e r m s in the last m e m b e r of (19), a n d F ( 0 ) w i l l in all cases be given b y (21), so t h a t we finally obtain
B y (8) we h a v e ( k + 4 ) f l < ~ , so t h a t the last m e m b e r tends to zero as u - - - ~ , a n d the l e m m a is proved.
6. P r o o f o f the case j = 1 o f the theorem
B y (18),
p=.P{er}
is defined as the p r o b a b i l i t y t h a t a t least one of the r a n d o m variables ~(0), ~(q), ~(2q) . . .~(mlq )
takes a value exceeding u. According to (17), this differs f r o m the p r o b a b i l i t yP{dr}
of a t least one ~q upcrossing in the first tl-interval b y a q u a n t i t y of the orderm c~tf~I~, Stationary stochastic process and high level B y L e m m a 3 t h e l a t t e r p r o b a b i l i t y is
P { d r } = q + o ( q ) .
T h u s we o b t a i n f r o m L e m m a 5, o b s e r v i n g t h a t b y (10) we h a v e 1/ue -u'/2 =o(q), P { E ~ ) ( k ) [ q + ~ 1 7 6 +C'
B y (10) we h a v e nq >~ a n d t h u s
li+m P { E k } = ~ e .
L e m m a s 1 a n d 4 t h e n finally give the relation (6) t h a t was to b e p r o v e d : lim Pk = ~ e-~.
U--> Oo k.[
T h u s we h a v e p r o v e d t h e simplest case of the t h e o r e m , w h e n i = 1, so t h a t t h e r e is o n l y one i n t e r v a l .
7. P r o o f o f the general case
T h e generalization to t h e case of a n a r b i t r a r y n u m b e r ~ > 1 of intervals is n o w simple.
F o r a n y ~ > 0, it follows f r o m the result just p r o v e d t h a t , for e v e r y i = 1 . . . ], t h e r a n d o m v a r i a b l e
N (a, + ~/~,bi - ~/tz),
where b , - a, = T,/~, will be a s y m p t o t i c a l l y Poisson d i s t r i b u t e d with p a r a m e t e r T , - 2 e . I n t h e s a m e w a y as in the proof of L e m m a 5 it is shown t h a t these ] v a r i a b l e s are a s y m p t o t i c a l l y i n d e p e n d e n t , so t h a t we h a v e
J (v, - 2 e)~*
P { N(a, + ~/[~, b, - ~/#) = k, for i = 1 , . , ?" } -+ 1-[ e (**-2,).
"" ~=1 kt! (26)
F r o m t h e a s y m p t o t i c Poisson distributions of the v a r i a b l e s N ( a , a ~ + s / # ) a n d N(b~-e/~,b~)
it f u r t h e r follows t h a t , with a p r o b a b i l i t y exceeding 1 - 27"e, these v a r i a b l e s will u l t i m a t e l y be zero for all i = 1 , . . . , j . Since 1" is fixed, a n d s > 0 i s a r b i t r a r i l y small, t h e t r u t h of t h e t h e o r e m t h e n follows f r o m (26).
ARKIV FOR MATEMATIK. B d 6 n r 2 0
R E F E R E N C E S
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3. CRAMs It., A" limit theorem~for t h e m a x i m u m values of certain stochastic processes. Teor.
Verojatnost. i Primenen. 10, 137 {1965).
4. CRim~R, H., a n d : L E i D B ~ g , M. R., Stationary a n d Related Stochastic Processes. To be published b y Wiley a n d Sons, New York.
5. K~INTCHINE, A. 52:., Mathematical Methods in the Theory of Queueing. Griffin a n d Co., Lon- don 1960.
6. I4OL~OGOROV, A. N., a n d ROZiNOV, Yu. A., On strong mixing conditions for stationary Gaus- sian processes. Teor. Verojatnost. i Primenen. 5, 222 (1960).
7. RICE, S. O., Distribution of the duration of fades in radio transmission. Bell Syst. Techn.
Journ. 37, 581 (1958).
8. VoL]~oNsxIz, V. A., a n d ROZANOV, Yu. A., Some limit theorems for r a n d o m functions. Teor.
Verojatnost. i Primenen. d, 186 (1959) a n d 6, 202 (1961).
Tryckt den 4 januari 1966
Uppsala 1966. Ahnqvist & Wiksells Boktryckeri AB
