View of On the level-upcrossings of stochastic processes

MATHEMATICÀ

-

^REVUE D'ANAI,YSE NUIUENTQUE

ET

^DE îHÉORIE DD IT''A.PPROXIM.q.TION

L'ANALYSB

NUMÉRIOUB

ET LA

THBORTB

I}E

L'APPROXIMATION Tome B, No

l,

^1979,

pp. 127-136

ON THE LEVEL-UPCROSSINGS _OF STOCHASTIC

PROCESSES

try

T¡AR¡.G ABDEI]

-

SAI,ÀM,q.TTIA (Kuwait)

ability

of at Teast ?t,

integrals.

upcrossing and

the next

one.is consi-

n-th

moment as

a double

integral

rst

occurrence. density

is

obtained

for a

general stochaãtic

for an

ergodic process

the n-th

momen-t

is

expressed

in

-2) ^nd

^moment

^{of the} probability of no

crossiägs

in

ttre

l.

Xntroduction

babilities of

some random variables ixed level

by

random processes

is

of

SEY l2l).

^For example the

om loading (ravrnoe

_[5]), e a few

of the many

areas _-

2.

Some Fundarnental Ilelationships

r,et

x(t) be a continuorls random variable. rre¡e we shalr consider some random variables associated

with

the crossings of x(t)

with

the level lø(l)l

: : L in its

duration

T. r.et E,(t)

be

the eveit tnal'lx(t)l

exceeds the'Íevel

L (n l)

times

in

the

i¡terval"(O, fl

^anð.the

^n-ti

"p[tàÁri"g-of

^{the rever}

i

occtlrs

at

time ú

in

the

horizontal

window sense of

xac

"rtõ

^{sr,EpraN l4).}

Define

the probability

densitíes

þ,(t)

^anð.

^{j^1t)}

^{as follows:}

þ"(t)dt

: plE"(t)

_:

_lø(0)lç Zl

128 and {2.1)

FARAG ABDEL _ _SALAM ATTIA 2

!

^{3 ,ON TIIE} LEVEL-UPCROSSINGS OF STOCIIASTIC ^PROCESSES 729

þ^ç¡at: ^PlE,(t)

^:

lø(o)l> Ll.

The probabilitylof at most ø

upcrossings

in

(0, ú)

is given by v,ilt) : _É u-pl

Êor

n : l,

_þLU) is the

first

occurrence density considered

by nrCI

anil È:0

¡pnn [7]'

The

probability of at

least ø upcrossings

of

the level

Z in

the

iltcrval (0, T) conditional to _lø(0)lç I is thus given by

T'T

'\

_Þ^lt)Or.

Similarly

\

^{O"Vl d,t}

is

^tine correspond.ing conditional

probability given

00 _lø(0)l

> ^{Z. Thus} the probability of at

least

n

upcrossings

in

the

'interval (0, 1) is

given

by

which by substitution from

equation (2.3) woulcl give

q.o 1

, 0

I

þr^r)d,t

n:0

\2.5) ^v,(t):

þ,+,(,)d.. + ^ã, _\ õ,þ)d'l

, -{o, ); ^{n2 |}

I

N

T T

I

0

':(2.2) ^a.o

þ,(t)dt +

^{ão þ,ç¡at,}

3. An

Inclusion-Exelusion

Formula lot þ"(t)

^{and, þ"!t)}

0

'where

ao:

^P tlø(O)l

( Il ^and ão: ^Pllx(O)l >

^¿1.

If

the event _tlø(O)l

< ^Zl

^is consid.ered.

to

be th-e

first

upcrossing _{_of_the} level

L,

^tlne probabilidy of exactly ^ø upcrossings

in

(0,

l) is

represented.

by

Ilere

we shall use a technique developed-

by ¡,c.nrr,Eg ill ^to

^obtain

a

representation

for

_þ,,(t) ^and.

f^1t¡ in

^terms

of

^certain

^multiple

^integrals'

Divide the interval

^(0,

l) into m eqtal

subintervals

Â1,

_{42, ' ' '}

', A'

¿nd define the

two

eventd e¡ ^{'anð, ø-¡}

to

d.enote an upcrossing

or no

upcfo- ssing

of the

^1eve1

L ín

L¿ respectively.

'Iake A;, i:1,2,....,m

^{so small -as}

^only

one crossing coulcl

takf

place,

iÎ

any,

in 4,.

Hence

if

_Q,þn) represents

the event that lhe n-th

äpctóssing

át tfr" lävel

_lø(/)l

: i'takerplace in the interval L^,

^m

^y'

^n,

then Q,(m) can be

written

as

the

Union

ZtT-):^utually

exclusive events (m-n)l

,

of the

from

U,(r) : ar{

I

^{ft t)d'r}

-

0

þ^¡1þ)d'r

+

0

{2.3)

+ã '{

0

þn-I ^{4 d.r}

0

þ^{,i r)d"c}

n2- | For ø:0

U o(t)

:

^au

{ 1-

_þ,(r)d't ^let

^f)

^ê,

^{(\ ..} f)ê¡,-tÀer,Uè¡+tÀ...n ë¡,-tÀe¡,1)

^èi,¡r

n "'

À ^ã¡o--t )

^e¡n-,À

^ë¿n-+rn

^{. .}

^. À

^ër,-t

I

^e-1.

(2.4)

Notice

that equation

(2.3) gives

lor n : I

^'Thus

tt

uLQ)

: o,{\ _nk)a' - _\ o,k)d,}* o,{, -\õ,þ)d'rl,

000

fr-fr+L ,n-tr+z i:I

UU

¿r:iL+l

lt-l

(3.1)

PlQ"(m)l:

P

U

i¡¡-1:i6-1},L

by the

assunption

that ilø(O)l> Ll is the first

upcrossing.

Equation (2.3) is

a

generalization

of

a formula given

by

nrcE and ¡EEe

[7] foi the

pròbability

bf failure of a

mechanical system subjected.

to

^a

random loading.

(êrf) ern .'. I ê¡-, O ¿;,lì e¡,+t)'.' _n

^è¡"-t

I ^{e¡,À ë¡,+tÀ} I ^ê¡,--tl) e¡,-l ^ë¡n-,+tn' "' À ã^-lÀ

^{e^)}

2 -' L'analvse luryéri-que ç! le !þéglie ^qq l'cppTsìri4ation : Tomc I' ^No' 2' 1979

FARAG ABDEL _ SALAM ATTIA 5 ON THE LEVEL-UPCROSSINGS OF STOCFIASTIC ^PROCESSES 131

130 ⁴

feplacing ^ë¡

by

¹

- ^{e¡ and}

^{after simple} manipulations the last formula ^is.

written in the

^form

using equation

(3.5),

the

conditional

probability

of at least ø upcros-

sings-in ih" iot"trr"l'(0,'l)

^given

that lr(0)l< I is

represented by, (3.2)

n, -ñ+l m-¡*2 t!-l' ( I l"^1 \ I

PlQ,@)l : _{,Ð ,,à,}

^'

^.' ,,_,-D-,*, l" LlA, _"',l À'^l-

-8,"[(Ö _',,)n'-f ⁺ f, ,fi*,"[ffi "r)n'-]- ^|

Taking the limit

^{as A,}

*

0,

þ,(t) ^is written in the

form

þ^(t) :lim PlQ"@):

|

*(0)l( Zl :

(a.s) :\at,\ot,'

,tt

_I{ Í"(t,,t,,.'.',t,-t,t)-

0 lr tn-

(3 6)

io.n o,: å

^t

-r)t-tl

^ln

^{*'i -2\} "_l ^,J"

0

I

,r

I

\ ^Í,t,,,,

t¡ *t-3

æ

'-'( ^n+x -

n-l ,)

X d'¡ dt, dt,

dtt

¡ ^t n+i-z¡ ^{r) d,t,}

* r_.r:D

t

- ^tl

^X

X

I

tr

dt,

f

lnìi J

-s

dtn*n , It¡i-2

Similarly

þb ^I d,¡

@/

)- ^{t-1)'-'l nti-2} _n-l

^X

æ

Ð

h=o

1)'l

^dt

t

I

ta

dtn-, .f"(tr,

tr,

. .

.,

^{tn+h)dtr+h (3.7)}

lr

fl I

\

0

f"(tr, tr,

^{. .}

., t,*¡-r,

t)d,t

Ir*h-l X dt,

dtr.",

where (3 4)

tn+i ₃

.f"(tr,

tr,

^. . .

.,

tþ)dt\, dtz h t-s'',';"

dto

:

4. Intervals

betwccn UPcrossings P

O {l'(¿,)l< ^¿ n

^lx(t,

I ^{dt)} r} : lr(o)l< ^z

(4.1)

^W(t)

: Mt + O(t) ast*

^0.

Using equation (2.5), u(t)

^can

^{be written in the}

^form

i.e.

.f,(tr,tr,..., tk)dtþtz...

dtL

is the

conditional

probability of

upcros-

.iogÉ"'iir ii,, rr'+"llt)',

^{(tr, t"}

¡"dtr),

^{. . . . , (tu, tr}

-l

dtuj

given that

_lr(0)l <

<¿.

The

independent variables

in f, ^{(tr, tr,}

^{' ' '}

:;

^to)..

t?! ^be

interchanged

arbitiarily

provia"d

-tttit

ip¡ fias'iÈË-

sáme

distiibution throughout

^its

time

durátiôn. Thus equation'(3.3) can be

written in

the form

(3.s)

þ,(tt)

:å _r-r);-,lø :'- r')5.,idt,

^{. . .}

,,j,_!^,,,t,,

. .

.,t¡ti-z,t)dtn+.-z'

- ^A

^similar expression torþ^1t7 can be obtained

where/, is

^{replaced by}

7, ^{and Í,(tr.,} tr,...,..,th)dtltz...,,dlo is jle conditional

probability

or upcros ,,og, ,n ^pr,

^-í,

t aq¡, (tr,ir l

^dlr), ' ' '

',

^{(to, th}

+ ^dth)

^giverr

that

_lø(0)l

<

^Z.

(4.2)

æ

u,(t)

:f

U ø:1

.,

"

:

å" [t - _{'. \Þ-*'?)a,} ^a,\o,oa]

Lat ^U(tt,

^t2),

^D(ü, t2) and C(tt, lr) Þ" the ^number o[

upcrossings, rlou,rrcrossingï

ã"a

cióssiãgs

of thclevcl I be x(t) in

(1,,

lr)

rcspectively.

Define, for t > ^0, h,:0, 1,2, '..

^'

(4

3)

^Rr(r,

^t) : PIU(-r,O)> I,

C(0,

r) <

^Àl

Ï32 and

i4.4)

^S¿(t,

^l) : PIU(-I,0)> ^{7, r(0)} > ^U, C(0,1)< /tl'

Thus Ro(t, /)

-

^So(r,4

^{< PIU(-r, 0)>} l,D(-.,0)>

I

< P lC(-r,0)>21.

Ry the regularity of the

stream

of

crossings,

the last inequality

gives

{4.5)

^Roþ,

^t) -

^{So(", l)}

^<

^O(r).

Ì{ow

So(tr+ rr, t): PIU(-rr,0)> l, f(0) > ^{U, C(0,4<}

^i¿l

+ t PIU(-tt-12, -'it) ^)- l, U(-.r,O) :0, x(0)> U, C(0,1)<

ft], The event

in the

second term on the

right

irnplies

that

there

is no

upcros- sing or downcrossin

in (-t, 0)

and

thus

also

x(-rt))

^{U, t.e.}

it

iruplies the

event lU(-", -r2, -"çL)) l, x(-tr)> U,

C(0, _Ð

<

å]

and

by

stationary

this is the

same ^as

lU(-rr, ^0)> l,

^*(0)

> ^U,

^C(0,1

f

^cr)

(

h l.

Thus

So(",

+ ^rr,t)?

^Su(tr,

^l) f

^S¿(rr,

t

J_¡t)

(

S¿(.r,

t) {

^Suþr,t).

Using a lemma

proved by rrrrNtcurNp [6],

.since So(r,

.l) ^is

^nondecrea-

sing as

r

increases, Sr(", l)/c converges

to

a

limit

as

c*

0. Thus,

lrom

eclua-

tlon

(4.5), koft, t)l,c also converges

to a limit

as

r+0

and hence

by

^(a.1)

Roþ,

t)lu(r)

tends

to

a

limit

as

r+0. ^{'lhis limit} isfinite,

since k,,ft,

t)lu(t)

5 l,

and we write

(4.6) zh(t):t::,w,

Zo

(t)

reptesents

the

conditional

probability of no

more

than å

crossings

in the interval

(0,

l), given that an

upcrossing occured

at t :0.

Define

now

(4.7)'

Fo(t)

: | -

^zþ_L(t),

h: t,2, ...

4'(4

smg f.unc

the next

upcrossing

is

represented.

by

Fr(l)

__ By definition, U"(t) is the probability of

rz upcrossings

in

(0,

t),

i.e.

U*tt)

:

PIU(Q,,

t).:

"1.

7 oN THE LEVEL_UPCUOSSTNGS OF STOCHASTIC PROCESSES

'Io

express F

r(t) in terms of

^U o(t)

,

^{we have}

^for

^,u

>

^{0 :}

Uo(t)

-

^Uo(t

* ^t) : ^plU(O, t) :01 - ^p l\(_r, ^t) : 0l : : _{PLU(-¡, 0)>7, U(0,1) :01} : : Rr(", t) +

^O(r).

Thus by

equations (a.1)

and

(a.6)

FARAG AI]DEL SALAM ATTI,\ ₆ 13fi

(4.8)

i*5ftlgdt ^{: _MZr(t)}

and hence.by using equation (4.7), tl,'e right-hand derivative D+Uo(t) exists

and

satisfies

(4.9) F2(t):t + ^M-rD+Uoþ).

5.

Moments

of thc

trnterval befwecn Uperossings

The

mean

of Fr(l) is

given

by

(5.1) Lt

- ^F,(t)ldt,

where

both

members ruay

be infinite.

Therefore (5.2)

Since

_{lU oQ}

*

^a)

_

U o(t)l

(

zøflcl),

it

follows.

that u

oþ)

is a

continuous

function of /

and hence

the

mean ^,of

Fr(l) is

given by

\,oo,p¡:

₀

+

^[1

^-uo(oo)].

By using relation

(2.S), we have

(t) M

j

tdF

0

\tan,ttl:

₀₀

\

\nn,p¡: - + \o*u,p¡at

0O

l, - ^{o,{ r} - _\ø,uto,¡|.

0

FARAG ABDEL - ^{SALAM ATTIA o}

q ON THE LEVEL-UPCROSSINGS OF STOCHASTIC PROCESSES 135

t3+

For an

ergodic process,

þr(t)dt: ¹

^{and hence}

[5] ^{K a rn e d} a, H., On the þrobability d,istri,bøtiotø of the number of crossings of ø cerløin resþonse leuel in raødotn uibratì,on. Mem. Fac, Þagrg. I(yoto Univ., 31rr 68, 83 (1e72).

[6] Khintchine Y. ^.A.., MatlternøticøI wethods iøthe theory of qweueing, Griffin, London, 1960.

[7] Rice, J.R. and Beer, F. P., First-occuvrence tivne o-f highJeuel crossings in ø con- tinuous ra'ndom þroaess, J, ^{Acoust, Soc. Amr., 39, 323-335 (1966).}

f

0

tdF2(,)

Un'iaersily of Kuwøit, Deþartment of Malhernølias

an

intuitively.reasonable ^result.

For

higher moments or F 2Q),

it is

straight forward

to

show

that

Received 15. V. 1976

(5.4)

Q.oI I

1- ^f

l

0

þ'

^{,Ç d'¡}

_-

^a.o

1-

t n'ola"llth-2d't:

_n(n - ^l)

æ@

o. [[ --tJ

¡n-zPr(r)d,rd't.

M 00

Again

for

an ergodic process, ^tine n

th

moment ^{0L F2(t)} is given by

(,5.5)

0

i

æ æ

I

I

n(n

-

¹⁾

t"dF_z\t) ^ø(n - ^7)ao

J *-'[t - ^þ,

^{þ)d"c d.t}

^:

^tn-2

^u

^ou)dt.

M M

0

The relations (5.4) and. (5.5) hold

in

the sense

that

^{both sides are either}

finite and

^equal

oi bôth

^infinite.

REFDRÞNCES

[l] ^B artlett, M' S., An inþod,uation to stochastic þfocesses. Cambriclge University ^Press, I,ondon, 1960.

t2]Blake,f.F.andI,ind.sey,Y'C',Leuel-crossingþroblemsforrandomþrocesses' IEE ^Trans. Inf. ^Theory, 11-19' ^{295-315 (1973)'}

t3lÇ¡amer,II.and.Leadbetter,M'F'',StøtionaryøndRclatedStochasliaProcesses' John WileY ^{ancl Sons,} New York' 1967

tA] Kac, M. ^antl Sllep ian', ^{D., L-ørge} excttrsions of Gøussøiøn þfoaesses' Ann' Math' Sta-

tist., 30, ^{1215-1228 (1959).}