A VERSION OF THE KRONECKER LEMMA

Size: px

Start display at page:

Download "A VERSION OF THE KRONECKER LEMMA"

Lorraine Pearson
5 years ago
Views:

1 UPB Sci Bull, Series A, Vol 70, No 2, 2008 ISSN 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

2 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

3 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 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

4 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 The coditios for Theorem re fulfilled Reltio (`) is lso stisfied: 2 lim 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

5 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

6 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)

7 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

8 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 , (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), North-Holld

Certain sufficient conditions on N, p n, q n k summability of orthogonal series

Certain sufficient conditions on N, p n, q n k summability of orthogonal series Avilble olie t www.tjs.com J. Nolier Sci. Appl. 7 014, 7 77 Reserch Article Certi sufficiet coditios o N, p, k summbility of orthogol series Xhevt Z. Krsiqi Deprtmet of Mthemtics d Iformtics, Fculty of