Strong Invariance Principle for Randomly Stopped Stochastic

Processes

Andrii Andrusiv, Nadiia Zinchenko

Introduction

Let be sequence of non-negative i.i.d.r.v. with d.f. and ch.f. ,

Denote

, , ,

where [a] is entire of a>0.

Let be sequence of non-negative i.i.d.r.v. independent of with d.f. and ch.f. ,

Denote

, ,

and define the renewal counting process as

n

ii

XnS1

)( 0)0( S ])([)( zSzS

)1:( iXi

F mEX

i

n

ii

ZnZ1

)( 0)0( Z ])([)( aZaZ

})(:0inf{)( txZxtN

)1:( iZi

)1:( iXi 1

F 1 1

iEZ

Main aim

The main aim of this talk is to study the asymptotic behavior of the random processes S(N(t)) and N(t) when F and F1 are heavy tailed. This problem has a deep relation with investigations of risk process U(t) and approximation of ruin probabilities in Sparre Anderssen collective risk model

)(

1

)(tN

kk

XctutU

)(

1

)(~

1

)(tN

ii

tN

kk

XVutU

Weak invariance principle

Limit theorems for risk process such as (weak) invariance principle which constitute the weak convergence of U(t) to the Wiener process W(t) with drift (when , ) or to α-stable Lévy process (when , ) lead to useful approximation of the ruin probability as a distribution of infimum of the Wiener process (Iglehard (1969), Grandell (1991), Embrechts, Klüppelberg and Mikosch (1997)) or infimum of the corresponding α-stable process (Furrer, Michna and Weron (1997), Furrer (1998)).

2

iEZ2

iEX

)(tY 2

iEZ2

iEX

Strong invariance principle

Strong invariance principle (almost sure approximation) is a general name for the class of limit theorems which ensure the possibility to construct and Lévy process ,

on the same probability space in such a way that with probability 1

as

as

were approximation error (rate) is non-random function depending only on assumption posed on .

)1:( iXi

)(tY 0t

))((|)()(| trotYmttS ))((|)()(| trOtYmttS

t

t

)(r

iX

Strong invariance principle for partial sums

Based on Skorokhod embeded scheme Strassen (1964) proved the first variant of the strong invariance principle. In 1970-95 the further investigations were carried out by a number of authors, so firstly I will summarize their results.

Th.A1. It is possible to construct partial sum process ,

and a standard Wiener process , in such a way that a.s.

,

with:

(i) iff , ,

(ii) iff ,

(iii) can be changed on iff

for some

)(tS 0t

)(tW 0t

))((|)()(| trotWmttS

pttr 1)( p

iXE || 2p

21))log(log()( tttr 2||i

XE))(( tro )(log))(( tOtrO iuXEe

0u

Domain of attraction of stable law

Suppose that ; more precisely we assume that

belongs to the domain of attraction of the stable law .

Here is a d.f. of the stable law with parameters ,

and ch.f.

where .

if for normalized and centered sums there is a weak convergence

2

iEX )1:( iX

i

,G

21 ,

G

1|| ))(exp()(,

uKug

))2tan()(sign1(||)()(,

uiuuKuK )()1:(

,GDNAiXi

*

nS

,

1* ))(( GmnnSnSn

Domain of attraction of stable law

Denote by , , the α-stable Lévy process with ch.f.

We omit index if it is not essential.

The fact that is not enough to obtain “good” error term above, thus, certain additional assumptions are needed. We formulate them in terms of ch.f.

))(exp();();(,,

utKutgutg

)()()(,

tYtYtY 0t

)()1:(,GDNAiX

i

Additional assumption

Assumption (C): there are , and such that for

where is a ch.f. of .

Put

01a 0

2a

luauguf |||)()(|2,

l

1|| au

)()( ueuf ium )(ii

EXX

]1)}()12(2),1([max{ lA

Strong invariance principle for partial sums

Th.A2. (Zinchenko) For and under assumption (C) it is possible to construct α-stable process , such that a.s.

,

for any

21 )(

,tY 0t

)(|)()(|sup 1

,0

TotYmttS

Tt

))1(41,0(),( Al

Counting renewal processes

Order of magnitude of N(t) is described by following theorem which includes strong law of large numbers (SLLN), Marcinkiewich-Zygmund SLLN and law of iterated logarithm for renewal process.

Counting renewal processes

Th.A3.

(i) If , then a.s.

(ii) if for some then a.s.

(iii) if then

while for the moments we have

,

10i

EZ

ttN )(p

iZE || )2,1(p

0))((1 ttNt p 2)(

iZVar

2321 |)(|)loglog2(suplim

ttNtt

t

ttEN ~)( 23~))(( tNVar

α-stable Lévy process

L.A1. If is an α-stable Lévy process with , then a.s.

Keeping in mind these and equivalence in weak convergence for Z(n) and associated N(t) it is natural to ask about a.s. approximation of N(t).

)(tY 20

)()( 1

totY0

Strong invariance principle for counting renewal processes

Under assumptions and strong approximation of the counting process N(t) associated with partial sum process

was investigated by a number of author. For instance, Csörgő, Horváth and Steinebach (1986) obtained that for non-negative r.v. the same error function (see T.A1) provide a.s. approximation

2EZ 10i

EZ

][

1

)(x

ii

ZxZ

iZ

))(())((|)()(| trOtrotWtNt

)(tr

Strong invariance principle for counting renewal processes

Let consider the case with and

Th.1. Let satisfy (C) with and

then a.s.

where is any upper function for Lévy process.

)(}{,GNDAZ

i

10i

EZ21

iZ 21 10

iEZ

))((|)()(|,

11 trotYtNt

)(tr

Strong invariance principle

Let recall

Strong invariance principle for D(t) was studied by Csörgő, Horváth, Steinbach, Deheuvels and other authors.

In the following we focus on the case when belong to , while can be attracted to the normal law ( , ) or to the

α2-stable law,

Our approach is close to the methods presented in Csörgő and Horváth (1993).

))(()( tNStD

2||i

XE )1:( iXi

)(,1 GDNA 21

1 )1:( iZ

i

2 2)( i

ZVar

212

Strong invariance principle

Th.2.(Zinchenko) Let satisfy (C) with and . Then a.s.

,for some

In this case D(t) can be interpreted as total claims until moment t in classic risk model.Developing such approach we proved rather general result concerning a.s. approximation of the randomly stopped process (not obligatory connected with the partial sum processes).

)1:( iXi

21 2

iEZ

)(|)()(| 11

,

totYtmtD ),0(01

),(

00l

Strong invariance principle

Let , be two real-valued positive increasing càdlàg random processes,

– the inverse of is defined by

,

)(* tZ )(* tS

)(* tN )(* tZ

})(:0inf{)( ** txZttN t0

Strong invariance principle

Th.3. Suppose that for some constants m, , and

functions , meet the conditions

, , , as

where is a Wiener process and being α-stable Lévy

process, independent of ,

Then

0a 0

))((|)())((|sup1

*1

0

TrOtWattZTt

)(1

tW

)(tr 0)(

ttr t

))((|)()(|sup *

0

TqOtYmttSTt

)(tY

)(1

tW

)(tq 0)(

ttq

0

at

Wa

mat

Ya

mttNS

2

** ))((

)))(log()()loglog(( 21)2(1 ttrtqttO

)(tr )(tq

Strong invariance principle

Th.4. Let satisfy (C) with and satisfy (C) with , . Then a.s.

for some

)()())(( 21

1

1

,

totYtmtNS

)1:( iXi

211 )1:( iZ

i

212 21

),(122

l

References1. Alex,M., Steinebach,J. Invariance principles for renewal processes and

some applications. Teor. Imovirnost. ta matem. Statyst., 50,(1994),22-54.2. Billingsley, P., Convergence of Probability Measures, J.Wiley, New York,

(1968).3. Csörgő,M., Horváth,M., Steinebach,J., Strong approximation for renewal

processes, C.R. Math. Rep. Acad. Sci. Canada.8,(1986), 151-154.4. Csörgő,M., Révész,P., Strong Approximation in Probability and Statistics,

J.Wiley, New York (1981).5. Csörgő,M., Horváth,L., Weighted Approximation in Probability and

Statistics, J.Wiley, New York (1993).6. Embrechts P., Klüppelberg C. and Mikosch T. Modelling extremal events:

for insurance and finance. – New York: Springer, 1997, 645p.7. Giknman I., Skorokhod.,A., The Theory of Stochastic Processes, II,

Nauka, Moscow (1973).8. Gut,A., Stopped Random Walks, Springer, Berlin (1988).9. Whitt, W., Stochastic-Processes Limits: An Introduction to Stochastic-

Process Limits and Their Application to Queues, Springer-Verlag, New York (2002).

10. Zinchenko,N., Strong Invariance Principle for Renewal and Randomly Stopped Processes, Theory of Stochastic Processes Vol. 13 (29), no.4, 2007, pp.233-245.

Thank you for attention!