UPB Sci Bull, Series A, Vol 70, No 2, 2008 ISSN 223-7027 A VERSION OF THE KRONECKER LEMMA Gheorghe BUDIANU I lucrre se prezit o vrit lemei lui Kroecer reltiv l siruri si serii de umere rele Rezulttele otiute se plic l studiul sirurilor de vriile letore I this wor it is preseted versio of Kroecer lemm cocerig rel umer series d sequeces The results otied re pplied to the study of rdom vrile sequeces Key words: Stolz-Cesro lemm, Kroecer lemm, rdom vrile sequeces Itroductio The Kroecer lemm cocerig rel umer series d sequeces is widely used i the field of Proilities i the study of rdom vrile sequeces The proofs of some theorems cocerig the lw of lrge umers d the lw of the iterted logrithm for sums of idepedet rdom vriles rely o the Kroecer lemm The theorem of RJTomis [2] tht estlishes reltio etwee the lw of the iterted logrithm d the lw of lrge umers is prove o the sis of this lemm I the pper of Gug-Hui Ci d Hg Wu [2] reltive to the lw of the iterted logrithm for sums of egtively ssocited rdom vriles, results re otied y employig the Kroecer lemm Kroecer lemm [9] hs the followig sttemet: if the rel umer series, is strictly icresig sequece hevig the is coverget d ( ) N limit lim, the it eists lim 0 () The pper wishes to preset versio of this lemm hvig s hypothesis the strictly decresig sequece ( ) N 2 A versio of the Kroecer lemm The mi purpose of this pper is to estlish the followig theorem Reder, Deprtmet of Mthemtics II, Uiversity Politehic of Buchrest, ROMANIA
38 Gheorghe Budiu Theorem Let ( ) e coverget rel umer series d,, N R N strictly decresig sequece If is coverget d lim 0, (2) ) the sequece ( ) N ) the sequece is coverget d N lim 0 (3) the the sequece is coverget d N lim 0 ( ) For the proof of the theorem it is ecessry the followig lemm which is versio of the Stolz-Cesro lemm Lemm Give the rel umer sequeces ( ) d ( y ) N N If ) the sequece ( y) is strictly decresig, N ) the sequeces ( ) d ( y ) N re coverget d lim lim 0 N y, c) there eists lim + l R y+ y, (4) the it eists lim l y (5) The proof of this lemm is well ow Proof of Theorem Give S, S 0, We hve +, 0 0,, S N
A versio of the Kroecer lemm 39 We trsform ( ), N (6) tig ito ccout tht S+ S: ( S S ) S ( S+ S) [ S + ( + ) S] + ( S S + S S + + S S + S S ) S 2 0 2 3 2 2 + Cosiderig tht 0 0, fter simplifictios we get: S+ S (7) We shll prove tht lim S S (8) Becuse lim S+ S, from reltio (7) it results reltio (`) To verify reltio (8) we shll use the lemm We must prove tht lim S 0 (9) For the clcultio of the limit we use the Ael trsform [5]: If A uv, V v the A uv ( u u ) V (0 + Itroducig u S, v i (0) d tig ito ccout (6) we hve: S S ( S S) S () + i i
40 Gheorghe Budiu The series eig coverget, the sequece of prtil sums ( S ) ouded, therefore lim S 0 By hypothesis lim 0, thus it results tht reltio (9) is fulfilled is To verify reltio (8) we pply the lemm The coditios ) d ) re give y hypothesis d y reltio (9) We verify coditio c) from the lemm: + S S + + lim lim lim S+ S + + S Reltio (8) hs ee verified d thus the theorem hs ee prove 2 Emple Give the coverget series d the sequece 3 2 ( ),, N N The sequece is strictly decresig d lim 0 It eists 2 lim lim 0 3 2 The coditios for Theorem re fulfilled Reltio (`) is lso stisfied: 2 lim 0 3 2 3 A Kroecer type limit Aother versio of the Kroecer lemm is give y the followig theorem: Theorem 2 Give the coverget rel umer series d the rel umer sequeces ( ) d ( ) If N N ) The sequece ( ) N ) The sequeces ( ), ( ) is strictly decresig, N N re coverget d lim lim 0, i i
A versio of the Kroecer lemm 4 c) It eists lim + + l R, the the sequece is coverget d i N lim 0 (2) Proof The sequeces ( ) d ( ) therefore it eists lim l N stisfy the coditios of the lemm i Let e S +, S 0, c, N,( 0 0) d S lim S N We trsform (2) hvig i mid the ove ottios: ( S+ S) ( S + S ) S + ( c + ) S d d [ ] [( S + S) cs ] S + cs S where + c S d d c S lim lim lim S l S + + + + But + + + By pplyig the lemm we get: + lim d lim S+ l Sl S 0 4 Applictio Limits of sums of rdom vriles I [9] is give d prove Theorem 3 Let e ( ) idepedet rdom vrile sequece hvig the N epecttio E ( ) 0, N i d ( g( ) ) fuctio sequece, eve d N i o-decresig for > 0 d tht stisfy oe of the followig coditios: ) the fuctio ( ) does ot decrese o the itervl (0, ) g
42 Gheorghe Budiu ) the fuctios g ( ) d g ( ) re ot icresig o the itervl (0, ) 2 If ( ) is coverget strictly positive umer sequece d if the series N E ( g ( )) g ( ) is coverget the the series (4) is coverget s (lmost sure) Tig ito ccout Theorem,we c chge the sttemet of Theorem 3 s followig: Theorem 3 Give the coditios of Theorem 3, if ( ) is coverget d N i strictly decresig positive umer sequece hvig lim 0, the, if lim 0 s, the it eists lim 0 s (5) Proof (If { Ω, K, P} is proility spce d : Ω R, N, the ( ( ω) ), for whtever ω Ω, (ω fied) is rel umer sequece, so the N ove results c e pplied to ( ( ω) ) o suset of proility of Ω ) N Applig theorem 3 it results tht the series is coverget (s) We verify the coditios of Theorem Coditio ) is give y hypothesis; for the verifictio of coditio ) we cosider, N i Theorem It results tht lim lim lim 0 s Thus coditio ) is stisfiedthe coclusio of theorem shows tht it eists lim lim lim 0, s which is ectly reltio (5) (3)
A versio of the Kroecer lemm 43 Corollry Let ( ) ( ) 0, i e idepedet rdom vrile sequece hvig N E N i d lim 0, s If ( ) is strictly decresig positive umer sequece hvig lim 0 N d the series p E, p 0< p < 2, (6) is coverget, the lim 0, s Proof The coclusio of the corollry is otied from Theorem 3 if we p cosider g( ) g( ) If 0 < p < the coditio ) from Theorem 3 is verified If < p < 2 the coditio ) from Theorem 3 is verified If p, the the reltios (3) d (4) re ideticl d we pply directly Theorem 3 Theorem 4 Let ( ) d ( ) ( ) g stisfyig the coditios of Theorem 3 If ( ) d ( ) N ) the sequece ( ) ) the sequece ( ) re rel umer sequeces fulfillig the coditios N is strictly decresig d lim 0 N N + c) it eist lim l R + is coverget d lim 0 ( ) ( ) d) the series E g g ( ) is coverget, the lim 0, s Proof: From Theorem 3 it results tht the series is coverget s
44 Gheorghe Budiu I Theorem 2 we cosider The coditios of Theorem 2 re fulfilled, thus lim lim lim 0, s R E F E R E N C E S [] B M Bud,SV, Fomi, Multiple Itegrls, Field Theory d Series Mir Moscou, 973 [2] Ci, GH, Wu, H,Lw of the iterted logrithm for NA sequeces with o idedicl distriutiosprocidi Acd SciVol 7,No 2,My 2007,pp23-28 [3] G Ciucu, Teori proilittilor si sttistic mtemtic Ed Tehic, Bucuresti,965 [4] I Cuculescu, Teori proilittilor Ed All, Bucuresti,998 [5] G M Fichteholz, Curs de clcul diferetil si itegrl,vol 2 Ed Tehic, Bucuresti,996 [6] CC Heyde, O lmost sure covergece for sums of idepedet rdom vriles, Shy, A30, No 4, pp 353-35, (968) [7] M Iosifescu, G Mihoc, R Theodorescu : Teori proilittilor si sttistic mtemtic, EdTehic, Bucuresti, 966 [8] M Loeve, Proility Theory, V Nostrd, New Yor,963 [9] V V Petrov, Summy ezvisimyh sluciyh veliciy Iz Nu, Mosv, 972 [0] V Petrov, Limit theorems of proility theory-sequece of idepedet rdom vriles, Oford SciecePulictio,995 [] D Pop, U criteriu petru clculul limitelor de sirurirmt, o 3, 997 [2] Tomis RJ, Refiemets of Kolmogorov s lw of the iterted logrithm Sttistics d Proility Letters 4 (992), 32-325 North-Holld