On the stability of the characterization of exponential distribution by geometric compounding

V N U J O U R N A L OF SCIE NCE Nat Sci t X V , n ° - 1999 O N T H E S T A B IL IT Y O F A C H A R A C T E R IZ A T IO N O F E X P O N E N T IA L D IS T R IB U T IO N B Y G E O M E T R IC C O M P O U N D IN G Tran K im Thanh Department o f M athem atic College o f Sciences, University o f Hue I IN T R O D U C T IO N Let X i , X , be nonnegative i.i.d.rv’s (independent identically distributed random variables) with F{ x ) = P { X j < x), ip{t) = m = E X j < +CX) and let V be independent of X j , j = 1,2 , with geomet,ric distribution, i.e., P { V = k) = k - , , ( < p < l , q - I - p) T h e random variable z = X ị + X ‘2 + ■■■+ X v is called geometric composed variable OĨ X f , T h e notation Gp(.r) will mean P { p Z < x); the n otation ự>pz{t) will m ean F q{x ) an d ^ o { f ) w ill d e n o te d is tr ib u tio n fu n c tio n a n d c h a r a c te r istic fu n c tio n r e sp e c tiv e ly o f th e ex po nential (listiilMition In [4], Renyi characterized the expo nen tia l distribution proving rh(' follovviiifi, assei tioiis; (i) /» n p -.o ỡ „(,r) = e - ''; (i i ) G',,(./■) = F ( r ) (Gp(,r)-= P { p z > x)), F ( t ) = e ~ ''; ( F ( x ) = - F( r }} In [2], we estim ated the stable (iegipp of this characterization in the case of the (listi ihutioii fuiiclioii F(.r) beiiifj f - ('xpoiipntial, i.e., T (f) > , T( f ) when f —> such that l + + + Y , / + ''/'■("//’) /'■("//>) + + + 11^ p- + II- 1 , - ^ ị = Ị p+ - í/ I íĩe r(í///í) |> p - p- + (Ị I ?ỉr r(////0 |> /í - (Ị.^ > V II :| // |< Ị) 'ỉ Conspquoiitly ' + '/í(Jpz("/p) i By (8) and V ' / :| í/ | < y).7’ (9) (9) we got | r(ỊỊ)| f \ p + (I^pz{!i/p) i T his completes the proof of Tiieoieni Ộ T h e o r e m As s um e tiiat Gp(:r) is f - exponential distribution fmiction with the niiinhei T ( f ) III ( l b ) satisfying the coiưlitivii T( f ) = { f “ ) for s o m e Í* > (when f —> 0) ĩ h e i i we hfive piF.Fo) < - ( l Oj h'je.luf, w h e r e < f < i n i n { l , } ỉìiici K ị > 0, ỉ aje c o n s ta n t niiinijers in d e p e n d e n t o f f Proof: At first, since Fo(.r) is exponential cliatiihution function tin'll sup I ) 1= L'siiif; E ssen ’s inequality (s(H* [3]) with T = />.T(f) \V(‘ i^('t the folluwiiii;