On the stochastic convergence of random vectors in real Hilbert space

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1 / 3 GAZETA DE MATEM ATIÇA wliere we suppose all with i < z already distributed as i *S_j_ aother sum is'^jj where the product ii 3+ i will appear; thia beig achieved by permutatig i 8 either ii =+ t+i ad a or. ii s+ t aud k + 1 factors of B. We remark here that with this operatio we ited to assomble i the same term, the elemets s, 1,. z+k, so the terms with less tha k + 2 factors will be already formed like i S+ ad will ot appear i ayoe of these terms. For this reaso B will be, i fact, a product of at least k + 1 factors. Let us iterchage the.+it+i with a. We get ad S-i = A z Ti^jt^s+i + B a + R jh ~ S = C^+fc+l a )(A-;.-. :+k B). As our aim is to obtai a sum JS'^-S 1, if Si S < 0 we iterchage -. with k +1 factors of B. I this case we obtai S 2 A N a + B' z :+k + R where B = N B' ad N is a product of k + 1 factors, ad we ca show that S 2 - S=( z - 7» :+k -N) ($' x +k+1 - A w B )>0. I fact we have : ;+k N ad o accout of the iequality A - = = NB> (implied by S { - S < 0), we get B' < A. Multiplyig this oe by < a we obtai B' s +jy.( < A a which proves the assertio. The other operatio is cocered with the fact that each term of j factors (for example s '.+i ca be costructed withi aotlier oe which may be a product of more tha j factors. As all are greater tha 1, if we iterchage the product ii; * with aother product of j factors as well, say, b ^, which is already a term of the iitial sum, we get a ew sum equal or greater tha the former. I symbols, from 8 = a b H + i : J+1 z+j - X + R where all with i < z are already distributed as i 8+ j we get S' =, =+ j_i-(- \-A a - -rih+r ad S { 8=( s - a ---h)(l A)>0 for - M =+j I < ad A > 1. By meas of this two operatios we ca get the sum from a arbitrary oe, say S\, through itermediate sums which will take sucessively oodecreasig values ad thus Theorem 2 is proved. REFERENCE [1] HARDY, LITTLE wo ON, PULYA. «iequalities». O the stochastic covergece of radom vectors i real Hilbert space por João Tig go Mexia 1. Itroductio The mai objectives of this paper are: i to obtai lower bouds of the probability of evets that are the itersectio of a deumerable or fiite family of evets, related each oe with a radom variable. ii to study the stochastic covergece of sequeces of radom vectors as arisig from coditios imposed o the sequeces of the compoets with the same idex. The case we are maily iterested i is whe the vectors have deumerable sets of compoets although we also cosider the case whe there is oly a fiite umber of compoets.

2 4 GAZETA DE MATEM ATIÇA To accomplish (í) we will joitly utilize the techique of passig to the complemetai ad probability iequalities of the TCÍIEBYCHEFF type amely the BIENAYMÉ- -TCHEBYCIIEFF ad PEARSON iequalities. To accomplish (it) we take the results pertaiig to (i) as a poit of departure ad utilize a techique itroduced by TIAGO DE OLIVEIRA [5], We shall begi with the study of the deumerable aalogue of the multiomial case, the we will geeralize our results to larger classes of radom variables ad, usig the same techique, obtai ew results. Afterwards ad through the same methods we will reach specific results cocerig the fiite case. We ed presetig cosistet estimators of the quatities itroduced. 2. A first case The results preseted i this sectio are cotaied i the laws of large umbers obtaied for rado elemets i Baach spaces by EDITH MOUKIER [3] but are derived through a much more elemetary techique. Let us start by obtaiig some fudametal iequalities; from ad é A m {A<) (1) (^4 -+ ii) ( (>4) ^ (B)) (3) I the fiite case : (4) -* ^ I j l ^ ^ l - S i ^ M ^ -. i V ). Let us cosider a probability space with a deumerable set of possible outcomes Wi with probabilities a^, ad let be the umber of times that i «tt» experieces we obtai <uo;s. We have i = l fiv V-i \»! -SW i=i \ tti Pi Ui Pi * = 1 b as follows from iequalities (2) ad from the BIENAYME-TCHERYCHEFF iequality: so that we obtai: (o)! i=1 ( \ tti " p i < e Let us study the stochastic limit of: 1 we 2 we have (2) ad prove: 2 tti Pi ^ 1 - f (A',) it is easy to give to (2) the followig form with a more geeral tur : PROPOSITION 1. 2 i!i ; = i Pi 0

3 GAZETA DE MATEM ÁTICA 5 ad P: Let us write :»(O-MÍfc x Ui 1 -s/4 ; 71 \ + oo 2 P' ~ ~~ > implies that the. 71 / ;=«( ) +! sum of the egative deviatios must be ^ -this fact joitly with 4 + CO 2 B i=a'(s) + l It's easy to see that A 7 (;) is fiite ad depeds oly o «SD ad 7 has mea value : M r 2 % i=tt( ) +1 < 4 implies that the sum of the positive deviatios must be <e/2. As we have ad : 2 Pi We also have ( \i=t Vri X Pi + 1 Pi < 4 iv(e) X 2 \ i=i t 2 < l k i - A O 4 / fil we see that r-/ i \ Pi < 1. ( 4iV(c)/ - i=l m g, ^ 2 $ + 1 < 7 -Pi < 4A 7 (0 which are cosequeces respectively from (5) ad from the BERNOULLI theorem. Let us observe that 2 f=i the absolute values of positive is the sum of deviatios ~>PiJ ad of egative deviatios: (') Let us suppose that the idexes «i«where classified, through classificatios C h j = 1, i disjoit subsets ad write:?(*,;= 2,2 vi P*,'~ 2 Pi Usig first icc. j i iec* at (5) ad the (4) we obtai ILMift^'il-')))! Y 1 > 1-2 ~ r m rte > TIE" From (4) ad the defiitio of A r («) follows f~ 00 L M 7H pi ad the, from (1), we get: (6) 2 V =1 71; which proves the desired 2 Pi < ~ ) i f i r - pi < 0 ^?i - -(4(iV(0)2 + l) result.

4 6 GAZETA DE MATEM ATIÇA Lei us defie ow the stochastic covergece of radom vectors i a HILHERT space by ^(s) = Mi 2 Pi ^ 1 \i-i 1 ~ e / 4 (7) (a a) if ad oly if: it is easy to see that (12) 1 J I s ) J N(s). where «= («, j), a~(a t ). We have As we see that q <i 1 we may have Pi > q ad whatever iv we may have (8). / - X 0 i = l V whose truth is verified easily by squarig both sides of the iequality. From (7) ad (8) follows pm> we see that it is impossible to obtai distributio free bouds for N(s) ad N(e). 3. Geeralizatio of the results (9) (1 Kj-W so we have (10) 1 -> ii ->p where»i,(ii ) ad p = (/>;) We also have, due to (1), (6) ad (5), the followig stroger result: We will ow cosider larger classes of a radom variables. The PEARSON iequality: (13) dtf-f.ks)^!--^ 8 r where» is the r th order absolute momet of the deviatios of «a;» from its mea value may be foud i SAVAGE [4]. From (3) ad (13) we get: W) w / 1=1 XI - 4 ( i + l ) ( 71 Observig (6) ad (11) we coclude the DO smaller JV(e) is the better. As 2 Pi = 1 i=l a series of o-egative terms we ca reor- 00 der it. Let 2 Pj = 1 J be the series after the j-1 terms havig bee reordered i order ad write is decreasig where «/3 r;> fb is the r* th order absolute momet of the deviatios of a?;» from it's mea aw;». Let's ow geeralize propositio 1. We have: PROPOSITION2. Let {A' }, with A; ([ =(IC.;), be a sequece of radom vectors i a real HILHERT space with o-egative (o- -positive) compoets, whose mea values «Pi» are idepedet of e». If, whatever 50 may be «KD, we have 2 x «,i S<C +

5 GAZETA DE MATEM ÁTICA ad: 2 r. - i w h e r e is the )--th order absolute momet of the deviatios of ffa,,,» from «.pi* we have a; ;i ->fjl where ft =i (ft,-) P: I KOUIOGOROV [1] it is prove CO that if the series 21 *! coverget, i = l «so the mea value of V i s ^ Pii where N* (e) depeds oly o <i t»; ad as if g (zj 8JT) = where ags is cotiuo us, ad [ V U (z? X z t )], the IT" = g (z'i,, Z'Y) Z u, MEXIA [2]; we have so that p,i = X* (f) + oo y I H 2 m = 2 Usig (14) we obtai + 00 y* - 2 t i=a'*(t) + l ^tr 0. «Î, is the mea of so that H= 2^''' We ca ow write is N* (E) = Mi 2 "I ^ # - E/4 - v Ui ' + «IV* (e) ad y = 2 2 Usie i =.V*(E>H-t (=1 The through the same techique that we used i provig propositio 1 we obtai: (15) ad 3%' JJ ^ I I X 1 y r (mm i=l / 11 -* CO 2 fc.'h) ftsbi from w a u d ( 8 we have: ) we see that as (i6) ^ ( «I B ^ ^ (!) Maitaiig the defiitios of C/,supposig oly that they are fiite, ad of p^ i ad writig 2 we obtai by usig first (14) ad i e Cj. i the $i (2) Co, I l i ^ f i ^ O )! > 1 - I v» v... Z^Mii I _> at oo Our thesis is ow proved. Observig (15) ad (16) we see that the smaller N~* (t) is the better, as before we ca itroduce: N*(<) = Mi 2 fv^ fi- s/4 where the defiitio of RB/ «is aalogues to that itiwll of where the <tfij are the a^» reordered i

6 8 GAZETA DE MATEM ATIÇA decreasig order. We see that: (17) 1 N*{t) N*{t) We will ow obtai a reslt related to propositio 2 but free from the restrictios we imposed o the sig ad mea value of the compoets: PROPOSITION Sf 1 ). Let ja;, with x = = {re,, fi), be a sequece of radom vectors i a real HILBERT space with meas aft«,' ' If there exist positive umbers uh,ii> such that for B = 2 QO 00 Jl,i, ad K = 2 if" PR ^2 <^1 ^T^ ^ i - fi puttig ri = ijjs we obtai (18) ^2 fé]i,i p»à<ij ^ K H,l * 1 r -f + 00 due to (1) ad (8) we the (19) y 2 - ^,0 2 < I ^ we have: II K X 1 * 1 E r -j- co the OO Ti -]- OO due to (1) ad the triagular property of metric we get: (20) ( 4 / 2 \ 2 <* + ad if there are umbers «a;» such that 2 ((*», (-«<)*-*<>, ^e f» «j where i=l a = («;). P: From (14) we have + tf?*" e oo as p (I a, i - fx ; 1 < v) h, i) j X A ^ ^ 1 J F r ill. t J h" rt r * t «ti f-1 «,i 'I [ - (*d < j - so that we see that out thesis is ow prove. opositios 2 ad 3 ca easily bo geeralized if istead of cosiderig oly momets of the r-th order we would cosider momets of orders We ow will prove: PROPOSITION 4('), Let A be a real HILiiEHT space; a cotiuous fuctio defi- ^ 2 I»,*" f*,i (i) If we admit that ago is i. e. the it ca be slioiv i the same way that: usig (1) we get (3) (as,,4x) -> (g ( ) $ g(x)).

7 X GAZETA DE MATEMATICA ed i A, aod, where x = (x,i), a sequece of radom vectors of A ad such that that x < 0 H{x) = 0 ; r > 0 - H(x) = 1»- a) There exists at A such that we get : {5 r>)lji = a<^ ^ we write where > d, a) -> 0, K,i = 2 / 3 \/~ - we get V i = I i= 1 V has as compoets tlie mea values of the compoets of x tl. b) There exist positive umbers «/i,;» for which I I K -+0. The g(x ]l )^g(a) II + GO D: Dae to the fact that «A is a real HILBERT space ad to (a) ad (b) we have: x ia, <tg beig cotiuous, for each >0, there exists vj( )>0 such that: - \ ad 1=1 so we have S r % j- co S f f i 7i co -* -j- oo so, due to (1) we have It's clear that the other coditio is satisfied. r( (6, f*{s))^i>4 g ) - ~ 9 I < ) so, as» ia, we see that our result is ow proved. We are ow goig to show by a example that there are sequeces for which the coditios of propositio 3 are satisfied. Let us take y.'j) j m- v/ r f$ o < ) 4. The fiite case We are ow goig to cosider results for the fiite case. Let arc] X\v be «A 7» radom variables with mea values «pi». From (4) ad (13) we get (f) ad as i=! ( t f. - f x. - I C ^ l - S =i +QC with 6'= 2W Si< ad<i $?(») L=1 s,icl] ( 2 fi( < 2?

8 10 GAZETA DE MATEM ÁTICA10 due to (1), we get For > A 7 we bave (21) (2 *(-M< 2 \i = l = 1 / i = 1 Let's itroduce iu R K d (a, b) = We have N the euclidea metric I. t«=i Pi < V -VW-l. I -V< ) x-i -V(E)-I 4 1 Ï-1 ) j-i j = l J so o accout of (1), (Ö) ad the trasitive property of implicatio we get w i «1 / it's easy to see, usig (4), tliat (22) q Oc ji X ai)l - where x ~ (a?, t ) ad a = (a ; ). (23) {mift * 1 Vw '-f io If us(ii)» is a fiite fuctio, we have: (24) i>lim(^(0 ~ ^(O)s() = 0 -j- oo so we see that the asymptotic distributio is a Heaveside distributio. Usig the same techique we obtai : 5. Eslimatio of N(E), N(S), ad N*(s) N \/ J Let us begi by the estimatio of N(e.). 71' We reorder the fractios i decreasig order :, )z,j,, ad write N* (0 = Mi Jg W*» (0 = Mi Y ^ 11 - «/4 - r V%} ad à i / im-i \ & t - t g») There exists N such that 1 > N -> _ < L. V where?,l >2r ad the are the aar ii» reordered i decreasig order. If s(w) is a fiite fuctio we obtai: (25) 0 = p lim ( m (e) - A)i a (») = = p lim (N* (i) - N* ( ) i() = = p lim (N (E) N* ( )) s (N) so that the asymptotic distributios are also the Fleavside oes.

9 GAZETA DE M ATEM ATIÇA 11 &i Refereces [1] KOLHOGOROY, A. N,, Foudatios of probability, Chelsea Publishig Compay, New York, C2] MEKÍÀ, J, T. P. N., Subsídios para uma teoria estatística do problema da classificação, Aais do Istituto Superior de Agroomia, Vol. 24. Lisboa, [3] MÍCBIEK. EOITH, Elemets aléatoires das u espace de Baach, Aales de l'istitut Heri Pomcaré, Vol. XIII, Parie, [4] SAVAGE RICHARD, obability iequalities of the Tchebycheff type, Joural of Research, series B, N. 3, 1961, Washigto, 1961, [5] TIAQO DE OLIVEIRA, J,, Estimação assitôíica de parâmetros quase lieares, Istituto Superior de Ciêcias Ecoómicas e Fiaceiras, Lisboa, I960, Sobre as várias maeiras de escrever as equações gerais da mecâica dos sistemas com um determiado úmero fiito de graus efe liberdade por P. de Varees e Medoça 1. Objectivo Ao publicar este artigo só um aspecto o osso ituito terá acaso excedido objectivos meramete didácticos o de chamar a ateção para a superioridade formal das equações de MIRA FERNAXDES(*) e de assim procurar fazê-las sair do esquecimeto em que ijustamete as matém aida a maioria dos programas uiversitários. 2. elimiar Cosideremos o sistema material C sujeito apeas a ligações bilaterais. Supohamos ser possível ecotrar um úmero u fiito de parâmetros (coordeadas gerais) q,(s = 1,2,, w) tais que todo o poto PeC é fução sòmete dos q a e do tempo í, uívoca e bidifereciável : (1) P = P(íi,? 2,---,? U) í)- Etão, o deslocameto virtual òp de P o istate t tem a expressão (2) 3P = (8 = 1,2,...,«). èq. (*) FBEMANDES, A. de MISA (1940) Equattioiíi delia Diamita. aportae Math.» 2: 1-6, e o seu deslocameto real dp o itervalo de tempo elemetar dí sucessivo ao istate t vale (3) dp=2-f P +, à à t Sejam as seguites as h < u equações de ligação (compatíveis e idepedetes) ão cosideradas quado da escolha dos u parâmetros q s (difereciadas quado holóomas): (4) 2?"d S s +,dí = 0 (* 1,2,-",A), s ode tato os y,., como os r> r são fuções de i e dos qt As equações (4) correspodem um deslocameto virtual (compatível) de C (5) 2 9» ò í, = 0 s O sistema C tem, por coseguite, k = u h graus de liberdade. Tirem-se de (4) os valores de h dos d q, por exemplo, os de dqk+\, d qk+2 > >àqlt e substituam -se em (3). Etão, estas equações covertem-se em *