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 giveq.o 1

, 0

### I

þr^{r)d,t}

### n:0

### \2.5) ^{v,(t):}

### þ,+,(,)d.. + ^{ã, } _{\ } õ,þ)d'l

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

### I

NT 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+llt-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

tadtn-, .f"(tr,

### tr,

. .### .,

^{tn+h)dtr+h}

^{(3.7)}

lr

fl I

### \

0

### f"(tr, tr,

^{. . }

### ., t,*¡-r,

t)d,tIr*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

tdF0

### \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

0tdF2(,)

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).}