J. Notn. Sci. Calm. Sri Lanka 1980 8 ( 1 )  : 31-34 

Conditionally Infinitesimal Systems of Random Variables 

K. L. D. GUNAWARDENA* 
Statistical Laboratory, Department of Mothenautics, Universizy of ~Uanchesfer, EngInrrd. 

(Paper accepted : 16 Jantcary 1980) 

Abstract : Infii~itestimal systems of random variables, defined for a sequence of 
' 

independent random variables are an important class of random variables in Probability 
Theory. In this papcr, conditionally infinitesimal systems are defined for a sequence of 
dependent random variables and some properties aro obtained. 

1. Introduction 

I.et { ( ) 1 k = 1, 2 ,......, k, ; n = 1 ,  2 ,...... be a double sequence of random 
variables which are row-wise independent. That is for rr = 1, 2...... ; k, -+ cx, as 
12 - ,. co 2nd for every n (which denotes the row) th: random variables X,,,, X,, ?... 

,Y,,,,, i;sc indcpcndent. 

Definition 1 . I  

A sequuncc of independent random variables ( ( X,,,, ) ) is said to be infinitesimal 
i f  for every sequence of integers k which satisfy I d k 6 k, for all n we have, 
x,,~ -t 0 in probability as n -+ oc. 

Notution. 
Y 

Xn , --.p 0 in probability as tz -t co will be written as X,, , - n -> a: 0. 

Consider it double sequence ( {  Xn,);)) k = 1 ,  2 ......, k,, ; n = 1, 2 ...... 
ratltlorn variablts. For every k ( l < k < kc,, ) and n = 1, 2 ,...... we have an 
inc rc i~~ i~ lg  soquence of cr -- fields F,," c F,,, c .... .. c l i , , k ,  such that every 
A',,,, is Fn., measurable. 

M70 sli~ll  cxlcnd the definiiion 1.1 to a sequencc of dcpcndcnt random varinb!cs 
and (tefinc "conclitionally irlfinilcstimal" as follows. 

Definition 1.2 

,Z sequence of rnndorn variables { X,,, ) ) is said to be conjitionally infinitcslhnal if 

fur  every E > 0. 
----- 

*Presently at tlre Department of hiathernatics, Univcrslty of Peradmiya, Peradeiuya. 



If the { ( Xn,, } ) are independent then it can be shown that the above definition is 
ecluivalent to definition 1 .1  and in the general case. implies it. [ 3 1 

2. Equivalent Definition and Properties. 
Lemma 2.1 

( ( X,,, ) ) is conditionally infinitesimal if and only if 

Proof : 

Let F ( x I Fn,,._, ) be a regular conditional distribution for Xn,, given Fn,,-,. [I] 

Taking rnax and first letting n -+ cx, and then a -+ 0 in (3) we get (2) if (1) 
l < k < k "  

Is true. 

E?. 
-- 

- - -  -. P ( I X,,, [ > E. I Fn,k-, ) 
1 + Q2 

Hence 

which implies (1)  if (2) is true. 

Note : If the ( {  Xn,k)) are independent then lemma 2.1 reduces to lemma 1,' 



l f  the double sequence ( ( X,,, ) ) of non-ncgative random variable: is co i~d i t i on~! [~  
iqfialitestimal then For ever)- t > 0. 

'I'l~eorem 2.2 is proved by taking max and first ldting n --+ a, and then 6 -<* 0 ;n (4). 
1 < k 6 I t "  

if' the double sequence ( { X,,,, ) ) is condit.ior~aily infinitcsimid then 

Yroof : 
30 

iLX 
\ E ( r  n l k . - l ~ F n , k - l ) ~ = = ~ / (  eitx - 1 ) ( (lx 1 l',,,k-.i j 1 

--CO 

The proof of the theorem is complcte by taking rnax first letting rl +w and then 
l d k d k ,  

E -+ 0 in (5 )  

Note : If the ( ( X,,,) ]r are independent theorem 2.3 reduces to Thcorcm 



3. Ger~eerdization of resdts in Section 2 Q d -dimension ( d  > 1 ). 

Let ( { XI,:, ) ) bt: a double array of Rd valued random vectors. 

A seqxence of rai~dom vectors { ( &,, ) ) is said tn be conditionally infinitesimal iT  
and only if 

for every E > O, where I ( x ( I = ( x I 2  1 xZ2 -I .. . .. . t xdZ) 4 for 3: =-- (xI, xy... ..., xd) 
E Rd. 

( ( A',,, ) ) is conditionally infinitesimal if and only i f  

If the double array { (Xn,, ) ) c IP, 4- d is co~~d~iionally infmitesii~r~rl tkcn for tvcry 
1 E K - t - f j  

Lemma 3.2 and theorem 3.3 are multidirneasional versions of 1cmm;t 2.1 and 
theorem 2.2 respectively and the proofs are sin2ilrrr to the proofs in one-dimensional 
case. 

I wish to thank Professor F. Papangelou for his Ixelg in the prcparaticrn of tbis 
manuscript and the as so cia ti or^ of Comlnor~vieaith Universities for providing a 
Conmonwcalth Scholarship. 

. BREIMAN, I,. (1968). PrababUity. Reading, Mass. Addision-Viosley Publishirlg Co. Inc.. 

2. G N ~ E N K O ,  B. V. & K O L M O ~ K O V ,  A. hi'. (1968). .Lirr.~it D~.vt:.ibr!tiot~s ,for St!rria of Ir:cl(<- 
petzdeilt Ranrlon~ VarifibllV. Revised edition, pg. 90, Rc:;t;il.tg. Mass. Additison-Vlesley 
Publishing Co. Inc., 

3.  LOEVE, M. (1968). Probability Tiieory, t?dr'd edition, pg 381. Princeton, 13. Van Nostiand Co. 
I I~c . ,  


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