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# View of On the level-upcrossings of stochastic processes

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(1)

MATHEMATICÀ

### -

REVUE D'ANAI,YSE NUIUENTQUE

### ET

DE îHÉORIE DD IT''A.PPROXIM.q.TION

NUMÉRIOUB

THBORTB

### I}E

L'APPROXIMATION Tome B, No

1979,

PROCESSES

try

T¡AR¡.G ABDEI]

### -

SAI,ÀM,q.TTIA (Kuwait)

of at Teast ?t,

integrals.

upcrossing and

one.is consi-

moment as

integral

### rst

occurrence. density

obtained

### for a

general stochaãtic

ergodic process

momen-t

expressed

moment

crossiägs

ttre

Xntroduction

### babilities of

some random variables ixed level

random processes

of

For example the

[5]), e a few

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)

the level lø(l)l

duration

be

### the eveit tnal'lx(t)l

exceeds the'Íevel

times

the

anð.the

the rever

occtlrs

time ú

the

window sense of

sr,EpraN l4).

Define

densitíes

anð.

as follows:

þ"(t)dt

:

### lø(0)lç Zl

(2)

128 and {2.1)

FARAG ABDEL _ SALAM ATTIA 2

### !

3 ,ON TIIE LEVEL-UPCROSSINGS OF STOCIIASTIC PROCESSES 729

:

upcrossings

(0, ú)

Êor

þLU) is the

### first

occurrence density considered

anil È:0

The

### probability of at

least ø upcrossings

the level

the

T'T

Þ^lt)Or.

O"Vl d,t

### is

tine correspond.ing conditional

00 lø(0)l

least

upcrossings

the

given

### which by substitution from

equation (2.3) woulcl give

q.o 1

, 0

þrr)d,t

N

T T

0

':(2.2) a.o

ão þ,ç¡at,

### 3. An

Inclusion-Exelusion

and, þ"!t)

0

'where

P tlø(O)l

¿1.

### If

the event tlø(O)l

is consid.ered.

be th-e

### first

upcrossing _of_the level

### L,

tlne probabilidy of exactly ø upcrossings

(0,

represented.

### Ilere

we shall use a technique developed-

obtain

representation

þ,,(t) and.

terms

certain

integrals'

(0,

subintervals

42, ' ' '

¿nd define the

### two

eventd e¡ 'anð, ø-¡

### to

d.enote an upcrossing

upcfo- ssing

1eve1

L¿ respectively.

so small -as

### only

one crossing coulcl

place,

any,

Hence

Q,þn) represents

äpctóssing

lø(/)l

m

### y'

n,

then Q,(m) can be

as

Union

### ZtT-):^utually

exclusive events (m-n)l

,

from

ft t)d'r

0

þ^¡1þ)d'r

0

{2.3)

0

þn-I 4 d.r

0

þ,i r)d"c

U o(t)

au

þ,(r)d't let

ê,

èi,¡r

e¡n-,À

. .

ër,-t

e-1.

(2.4)

Notice

(2.3) gives

'Thus

tt

uLQ)

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

000

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

¿r:iL+l

lt-l

(3.1)

P

i¡¡-1:i6-1},L

assunption

### that ilø(O)l> Ll is the first

upcrossing.

Equation (2.3) is

generalization

a formula given

nrcE and ¡EEe

pròbability

### bf failure of a

mechanical system subjected.

a

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

(3)

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

130 4

feplacing ë¡

1

### - e¡ and

after simple manipulations the last formula is.

form

(3.5),

conditional

### probability

of at least ø upcros-

given

### that lr(0)l< I is

represented by, (3.2)

n, -ñ+l m-¡*2 t!-l' ( I l"^1 \ I

'

as A,

0,

form

|

,tt

0 lr tn-

(3 6)

t

ln

0

,r

*t-3

æ

### n-l ,)

X d'¡ dt, dt,

dtt

¡ t n+i-z¡ r) d,t,

t

X

X

tr

dt,

lnìi J

-s

dtn*n , It¡i-2

Similarly

þb I d,¡

@/

X

æ

h=o

dt

t

ta

dtn-, .f"(tr,

. .

tn+h)dtr+h (3.7)

lr

fl I

0

. .

t)d,t

Ir*h-l X dt,

where (3 4)

tn+i 3

.f"(tr,

. . .

### .,

tþ)dt\, dtz h t-s'',';"

dto

### 4. Intervals

betwccn UPcrossings P

lx(t,

W(t)

0.

can

form

i.e.

dtL

conditional

upcros-

(tr, t"

### ¡"dtr),

. . . . , (tu, tr

dtuj

lr(0)l <

### The

independent variables

' ' '

to)..

interchanged

provia"d

sáme

its

### time

durátiôn. Thus equation'(3.3) can be

the form

(3.s)

þ,(tt)

. . .

. .

### - A

similar expression torþ^1t7 can be obtained

replaced by

probability

-í,

dlr), ' ' '

(to, th

giverr

lø(0)l

Z.

(4.2)

æ

u,(t)

U ø:1

"

t2),

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

upcrossings, rlou,rrcrossingï

cióssiãgs

(1,,

rcspectively.

'

(4

Rr(r,

C(0,

Àl

(4)

Ï32 and

S¿(t,

Thus Ro(t, /)

So(r,4

I

stream

crossings,

gives

Roþ,

So(", l)

O(r).

Ì{ow

i¿l

ft], The event

### in the

second term on the

irnplies

there

### is no

upcros- sing or downcrossin

and

also

U, t.e.

iruplies the

C(0, Ð

å]

and

stationary

same as

*(0)

C(0,1

cr)

h l.

Thus

So(",

Su(tr,

S¿(rr,

J_¡t)

S¿(.r,

Suþr,t).

Using a lemma

.since So(r,

nondecrea-

sing as

### r

increases, Sr(", l)/c converges

a

as

0. Thus,

eclua-

### tlon

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

as

and hence

(a.1)

Roþ,

tends

a

as

since k,,ft,

and we write

Zo

reptesents

conditional

more

crossings

(0,

### l), given that an

upcrossing occured

now

Fo(t)

zþ_L(t),

smg f.unc

upcrossing

represented.

Fr(l)

rz upcrossings

(0,

i.e.

U*tt)

PIU(Q,,

### t).:

"1.

7 oN THE LEVEL_UPCUOSSTNGS OF STOCHASTIC PROCESSES

express F

U o(t)

we have

,u

0 :

Uo(t)

Uo(t

O(r).

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

satisfies

Moments

### of thc

trnterval befwecn Uperossings

mean

given

(5.1) Lt

where

members ruay

Therefore (5.2)

lU oQ

a)

U o(t)l

zøflcl),

follows.

oþ)

continuous

and hence

mean ,of

given by

0

[1

(2.S), we have

(t) M

## j

tdF

0

00

0O

### l, - o,{ r - \ø,uto,¡|.

0

(5)

FARAG ABDEL - SALAM ATTIA o

q ON THE LEVEL-UPCROSSINGS OF STOCHASTIC PROCESSES 135

t3+

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),

straight forward

show

(5.4)

Q.oI I

0

d'¡

a.o

_n(n - l)

æ@

¡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

æ æ

I

I

n(n

### -

1)

t"dFz\t) ø(n - 7)ao

þ)d"c d.t

tn-2

### u

ou)dt.

M M

0

The relations (5.4) and. (5.5) hold

the sense

### that

both sides are either

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).

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