STRONG LAW OF LARGE NUMBERS FOR SCALAR-NORMED SUMS OF ELEMENTS OF REGRESSIVE SEQUENCES OF RANDOM VARIABLES

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1 Annales Univ Sci Budapest, Sect Cop 39 (2013) STRONG LAW OF LARGE NUMBERS FOR SCALAR-NORMED SUMS OF ELEMENTS OF REGRESSIVE SEQUENCES OF RANDOM VARIABLES MK Runovska (Kiev, Ukraine) Dedicated to the 70 th anniversary of Karl-Heinz Indlekofer Counicated by OI Klesov (Received Deceber 14, 2012; accepted Deceber 26, 2012) Abstract The necessary and sufficient conditions providing the strong law of large nubers for scalar-nored sus of eleents of first-order regressive sequences of rando variables with independent and syetric disturbance are studied 1 Introduction Consider a linear regressive sequence of rando variables (η k ) = (η k, k 1) which obeys the syste of following recurrence equations: (1) η 1 = β 1 θ 1, η k = α k η k 1 + β k θ k, k 2, Key words and phrases: Strong laws of large nubers for sus of rando variables, regressive and autoregressive sequences of rando variables, sus of independent rando eleents with operator noralization 2010 Matheatics Subject Classification: Priary 60G50, 65B10, 60G15; Secondary 40A05 This work is partially supported by DFG grant STA711/1-1

2 366 MK Runovska where (α k ) and (β k ) are nonrando real sequences, and (θ k ) is a sequence of independent syetric rando variables such that (2) P{θ k = 0} < 1, k 1 Reind that the rando variable θ is called syetric if θ and ( θ) are identically distributed Note that introduced recurrent schee involves a wide class of sequences For exaple, if (θ k ) is a standard Gaussian sequence then (η k ) is a Gaussian Markov sequence The partial case of sequence (1) is a sequence of independent rando variables, when α k = 0, k 2 It is well-known that in the classical works of P Levy [13], AYa Khintchine [11], AN Kologorov [12], YuV Prokhorov [17] and M Loeve [14] the theory of alost sure asyptotic behavior of increasing scalar-nored sus of independent rando variables was constructed This theory aong other probles studies different versions of the strong law of large nubers One of the ost classical types of the strong laws of large nubers is the Prokhorov-Loeve type strong law for scalar-nored sus of independent rando variables Such type of the strong law of large nubers was intensively studied in the works of VV Buldygin [1], AI Martikainen and VV Petrov [15, 16], NA Volodin and SV Nagaev [22] The ain purpose of this paper is to obtain general conditions providing strong law of large nubers for scalar-nored sus of eleents of sequence (1) siilar to the Prokhorov-Loeve type Naely, for the sequence (1) we will consider the sequence of partial sus (S n ), S n = n η k, n 1, and study conditions providing c n S n alost surely, where (c n ) is soe nonrando real sequence It is also worth noting that another partial case of sequence (1) is an autoregressive sequence of rando variables, when α k = q = const, k 1 In this case sequence (1) is a stationary sequence For wide sense stationary sequences of rando variables the strong law of large nubers was studied in the works of IN Verbitskaya [20, 21], and later VF Gaposhkin [8, 9, 10] We consider autoregressive sequence as an exaple which illustrates ain result of this paper The set proble is tightly connected with the proble on conditions for the alost sure convergence of series η k Necessary and sufficient conditions providing alost sure convergence of such a series, in particular Gaussian Markov series, were obtained in papers of VV Buldygin and MK Runovska [3, 4, 5, 18, 19] The technique applied by studying series η k is based on the passage fro the sequence of partial sus (S n ) of regressive sequence (1) to the sequence of sus of independent rando eleents in higher diensional space This technique is general and is effectively used also in this paper

3 Scalar-nored sus of eleents of regressive sequences Strong law of large nubers for scalar-nored sus of eleents of regressive sequences Let n 1 and 0, 1 n < k; 1, n = k; (3) a(n, k) = 1 + n k ( α j ), n > k, and A(n, k) = β k a(n, k), n, k 1 The following Theore gives necessary and sufficient conditions providing the strong law of large nubers for sus whose ters are eleents of regressive sequence (1) By R denote the class of all sequences of positive integers increasing to infinity Theore 21 Let S n = n η k, n 1, for the regressive sequence (1), and (c n ) be soe nonrando real sequence In order for the relation c n S n alost surely, to hold it is necessary and sufficient that the following two conditions be satisfied: 1) li c na(n, k) = 0, for any k 1; 2) for all the sequences ( j ) of the class R c j+1 k= j+1 A( j+1, k)θ k alost surely Preparatory to proving the Theore 21 we recall one auxiliary result which will be used to prove sufficiency of Theore 21 We interpret the space R d, d 1, as d-diensional real Euclidean space of colun vectors with the inner product X, Y and Euclidean nor X = = X, X, X, Y R d Let c 0 (R d ) be the space of all convergent to zero sequences of eleents of the space R d ; (X n ) be the sequence of independent syetric rando vectors in R d ; Ξ n = n X k, n 1; (A n ) be the sequence of continuous linear operators apping R d into R d The following Proposition is due to VV Buldygin and SA Solntsev (see [6, 7]) and provides a criterion for the alost sure convergence to zero of operatornored sus of independent syetric rando vectors in the space R d

4 368 MK Runovska Proposition 21 In order for the relation (A n Ξ n ) c 0 (R d ), alost surely, to hold, it is necessary and sufficient that the following two conditions be satisfied: A) (A n X k ) c 0 (R d ), alost surely, for any k 1; B) for all the sequences ( j ) of the class R A j+1 (Ξ j+1 Ξ j ) alost surely Proof of Theore 21 First let us prove the necessity of Theore 21 Consider the sequence (c n S n ) and represent it in the for of a series: c 1 S 1 c 1 A(1, 1) 0 0 c 2 S 2 c 2 A(2, 1) c 2 A(2, 2) = θ 1 + θ θ n +, c n S n c n A(n, 1) c n A(n, 2) c n A(n, n) whose ters are independent syetric rando eleents in the sequence space R Ephasize that this series is coordinate-wise convergent Let us rewrite the latter series as follows S = ( A k θ k ), where S = c 1 S 1 c 2 S 2 c n S n, 0 c 1 A(1, k) c 2 A(2, k) A k = = 0 c n A(n, k) c k A(k, k), k 1 c k+1 A(k + 1, k) Since (c n S n ) c 0 (R), alost surely,then A k θ k c 0 (R) alost surely, for any k 1 Indeed, let us represent the R -valued rando eleent S in the for of the su S = S 1 + S 2, where S 1 = A 1 θ 1, S 2 = k=2 A k θ k, and show that A 1 θ 1 c 0 (R), alost surely Since (θ k ) is a sequence of independent syetric rando variables, then eleents S 1 and S 2 are independent syetric eleents in the space R Hence eleents ( S 1 + S 2 ) and ( S 1 S 2 ) are identically distributed Since S c 0 (R), alost surely, ie ( S 1 + S 2 ) c 0 (R),

5 Scalar-nored sus of eleents of regressive sequences 369 alost surely, then one has that also ( S 1 S 2 ) c 0 (R), alost surely Taking into account the linearity of the space c 0 (R), we get that S 1 c 0 (R), alost surely, and S 2 c 0 (R), alost surely, that is A 1 θ 1 c 0 (R), alost surely, and k=2 A k θ k c 0 (R), alost surely By analogue, one can see that A k θ k c 0 (R), alost surely, for any k 1 The latter relation together with (2) iplies that A k c 0 (R), for any k 1, ie li c na(n, k) = 0, for any k 1 Thus, assuption 1) of Theore 21 holds To prove that assuption 2) of Theore 21 holds, we shall follow the contraction principle in the space of all convergent sequences (for ore details see [6, 7]) Consider the set of rando variables (w n,k ; w n,k = c n A(n, k)θ k, k, n 1 n, k 1), where This set possesses the following properties: a) for any n 1, the series w n,k alost surely converges in the space R; b) the sequences W k = (w n,k, n 1), k 1, are independent and syetric as eleents of the space of sequences R Moreover, n (4) c n S n = c n A(n, k)θ k = c n A(n, k)θ k = w n,k, for any n 1 Fix an arbitrary sequence ( j ) belonging to the class R Along with the set (w n,k ; n, k 1) consider the nonrando set (λ n,k ; n, k 1), where { 1, n = j+1, j < k j+1, j 1, λ n,k = 0, otherwise Obviously, for the set (λ n,k ; n, k 1) the following condition holds V ar(λ n,k ; n, k 1) = sup sup n 1 2 [( 1 ) ] λ n,k λ n,k+1 + λ n, = 2 < Hence (λ n,k ; n, k 1) is a contraction set of bounded variation (see [6, 7]) Besides, according to the definition of sets (w n,k ; n, k 1) and (λ n,k ; n, k 1) one has that for any k 1, λ n,k w n,k 0, alost surely

6 370 MK Runovska Further, since c n S n alost surely, then by virtue of (4) the sequence of rando variables ( w n,k, n 1) converges alost surely in the space R Thus, all the assuptions of the contraction principle in the space of convergent sequences (see [6, 7]) hold The latter iplies that λ n,k w n,k alost surely This relation in our ters ay be rewritten as follows: k= j+1 c j+1 A( j+1, k)θ k alost surely Thus, condition 2) of Theore 21 holds This copletes the proof of necessary part of Theore 21 Now let us focus on the sufficiency part of Theore 21 First note that the sequence of partial sus (S n ) obeys the syste of following second-order recurrence equations: (5) S 1 = S 0 = 0, S n = (1 + α n )S n 1 α n S n 2 + β n θ n, n 1 Now we pass fro the relations (5) to the first-order recurrence relations in the space R 2, naely (6) S1 = Θ 1, Sn = B n Sn 1 + Θ n, n 2, where S n = ( Sn S n 1 ), B n = ( ) 1 + αn α n, Θ 1 0 n = Now we transfor recurrence relation (6) to the following for ( 2 (7) Sn = j=n ( 3 B j )Θ 1 + j=n ( ) βn θ n, n 1 0 B j )Θ B n Θ n 1 + Θ n, n 1, where k j=n B j = B n B n 1 B k, k n Using the atheatical induction principle and (3) one ay obtain that k B j = j=n ( ) a(n, k 1) 1 a(n, k 1), 2 k n a(n 1, k 1) 1 a(n 1, k 1)

7 Scalar-nored sus of eleents of regressive sequences 371 Now we prove the sufficiency part of the theore We show that assuptions 1) 2) provide c n S n alost surely We distinguish between the following two cases: I) all coefficients (α k ) are nonzero, that is α k 0, k 2; II) there are soe zero coefficients aong (α k ) Case I) Let all coefficients (α k ) be nonzero, that is α k 0, k 2 Then all atrices B n, n 1, involved in representation (7) are non-singular, that is det B n 0, n 1 This eans that one can pass fro relation (7) to the following one ( 2 S n = j=n B j )( Θ 1 + B 1 2 Θ 2 + (B 1 2 B 1 3 )Θ (B 1 2 B 1 3 B 1 n )Θ n ), n 1, where B 1 k is the inverse atrix for B k, k 1 Now we represent the sequence ( S n ) as a sequence of sus of independent rando vectors with an operator noralization (see [6, 7]), naely where A 1 = ( ) 1 0, A 0 1 n = S n = A n n 2 B j = j=n Ξ n = X k = A n Ξ n, n 1, ( ) a(n, 1) 1 a(n, 1), n 2, a(n 1, 1) 1 a(n 1, 1) n X k, n 1, ( k X 1 = Θ 1, X k = j=2 ) B 1 j Θ k, k 2 Note that (X n ) is a sequence of independent syetric rando vectors in the space R 2 Thus, sequence ( S n ) is represented in the for of the sequence (A n Ξ n ) of sus of independent rando vectors with an operator noralization in R 2 For n 1 set C n = ( cn 0 0 c n 1 ),

8 372 MK Runovska and consider the sequence (C n A n Ξ n ) which is also the sequence of sus of independent syetric rando vectors in the space R 2 Therefore, c n S n alost surely, if (C n A n Ξ n ) c 0 (R 2 ), alost surely Now to coplete the proof of sufficiency part of the theore in case I) one should check assuptions A) and B) of Proposition 21 for the sequence (C n A n Ξ n ) Case II) Without loss of generality we will assue that there is an infinite nuber of zeros aong α k s, k 2 This iediately iplies that the liit A(n, k) exists for any k 1 Therefore, assuption 1) of Theore 21 li holds if and only if li c n = 0, whence sup n 1 c n < Note also that if in assuption 2) of Theore 21 we set j = j, j 1, then we obtain that li c jβ j θ j = 0 This iplies that sup c n β n θ n <, alost surely n 1 To prove the theore in case II) we apply the disturbed coefficients ethod siilar to that one introduced in [6, 7, 18] Naely, we will construct an auxiliary regressive sequence ( η k ) which would obey representation (1) involving nonzero coefficients α n, n 1, such that all the assuptions of Theore 21 hold for it, and cn (Ŝn S n ) alost surely, where Ŝn = n η k, n 1 Thus, along with sequence (1) consider a regressive sequence ( η k ) which obeys the syste of recurrence equations η 1 = β 1 θ 1, η k = α k η k 1 + β k θ k, k 2, where (β k ) and (θ k ) are the sae as in (1), while { α k, if α k 0, α k = ε k, if α k = 0 The sequence (ε k ) is constructed such that ε k > 0, (8) k 1, and α j 2 k 2 l c k, k, l 1 First we show that such a sequence (ε k ) exists We enuerate the indices k 1 such that α k = 0 The resulting sequence is denoted by (n i ), i 1, that is α ni = 0, i 1 Now we choose the sequence (ε ni ) such that ε ni 2 n ni (i+1) j=1 ĉ j, i 1, δ ni

9 Scalar-nored sus of eleents of regressive sequences 373 where and δ ni = ni = sup s n (i 1) +1,t n (i+1) 1 { 1, if ni (0, 1), ni, if ni [1, ), ĉ k = n i 1 =s α { 1, if c k > 1, c k, if c k 1 t =n i+1 α, i 1, Fix k 1, and note that α j = 0, l 1, if α j 0 for all k + 1 j k + l Thus we ay restrict the consideration to the case where there exists at least one nuber n i {k + 1, k + 2,, k + l} such that α ni = 0 Then α j = αj k+1 j :α j 0 c k i:k+1 n i α j = i:k+1 n i αj k+1 j :α j 0 2 n (i+1) ni j=1 ĉ j δ ni 2 n (i+1) 2 () c k, εni i:k+1 n i for all l 1 This eans that inequalities (8) hold if the sequence (ε ni ) is chosen as indicated above We continue the proof of Theore 21 for the case II) and show that assuptions 1)-2) of this theore hold for the sequence ( η k ) Let Â(n, k), n, k 1, be defined as A(n, k), n, k 1, with coefficients α k, k 2, instead of α k, k 2, involved Since ) ( n k cn ( c n (Â(n, k) A(n, k) = n k c n α j )β k ) n k βk α j c n 2 l 2 k c k β k = = c n (2 k 2 n ) c k β k 0,

10 374 MK Runovska for any k 1, then by virtue of assuption 1) of Theore 21, li cnâ(n, k) = = 0, for any k 1 Thus, assuption 1) of this Theore holds for the sequence ( η k ) Fix an arbitrary sequence ( j ) belonging to R Then c j+1 = c j+1 c j+1 c j+1 k= j+1 k= j+1 k= j+1 k= j+1 c j+1 = c j+1 (Â(j+1, k) A( j+1, k)) θ k = j+1 k j+1 k k= j+1 k= j+1 ( j+1 k α j )β k θ k α j βk θ k 2 k l c k β k θ k = j+1 k 2 k c k β k θ k 2 l 2 k c k β k θ k c j+1 sup c n β n θ n n 1 k= j+1 2 k 0 Hence according to assuption 2) of Theore 21, one has that assuption 2) of this theore also holds for the sequence ( η k ) Therefore the sequence ( η k ) satisfies all the assuptions of Theore 21, whence we conclude that c n Ŝ n alost surely, in view of the case I) proved above Finally, let us show that c n (Ŝn S n ) alost surely Indeed, c n ( n c n (Ŝn S n ) = c n k) (Â(n, k) A(n, k))θ n n k α j βk θ k c n n n k 2 k l c k β k θ k = n n = c n (2 k 2 n ) c k β k θ k c n 2 k c k β k θ k sup c k β k θ k c n 0 k 1

11 Scalar-nored sus of eleents of regressive sequences 375 Since c n Ŝ n 0, alost surely, we conclude that c n S n alost surely, too Thus, Theore 21 is valid for the case II), as well, and the proof is coplete Reark 21 Except extreely degenerate cases of the sequence (η k ), assuption 1) of Theore 21 ay be replaced by li c na(n, k) = 0, for any k 1 Reark 22 Assuption 2) of Theore 21 holds if and only if for all the sequences ( j ) of the class R and any ε > 0 j=1 P{ cj+1 k= j+1 A( j+1, k)θ k > ε } < Reark 23 In assuption 2) of Theore 21 we do not have to deand that the appropriate condition holds for all the sequences ( j ) belonging to class R Priarily the class R appeared in works [16, 1, 2] In particular, in these works assuptions siilar to assuption 2) of Theore 21 ay be found In [1, 2] was shown that the corresponding assuptions ay be restricted to involve only soe specially constructed so-called testing sequences instead of all sequences ( j ) of the class R The construction algorith of the explicit for of the testing sequences ay be found in [2] The following Corollary shows that the result of Theore 21 corresponds to the well-known Prokhorov-Loeve type strong law of large nubers for scalarnored sus of independent rando variables (see [16, 2]) Corollary 21 Let α k = 0, k 2, ie η k = β k θ k, k 1 Then c n S n 0, alost surely, if and only if li c n = 0, and for all the sequences ( j ) of the class R c j+1 k= j+1 β k θ k alost surely Let us consider soe exaples Exaple 21 Let c n = 1 n, n 1 Then according to Theore 21 1 n (η 1 + η η n ) 0, alost surely, if and only if the following two conditions hold:

12 376 MK Runovska 1 n 1) li A(n, k) = 0, for any k 1; 2) for all the sequences ( j ) of the class R 1 j+1 k= j+1 A( j+1, k)θ k alost surely Exaple 22 Consider an autoregressive sequence (η k ): η 1 = β 1 θ 1, η k = qη k 1 + β k θ k, k 2, where q is soe real constant Let us distinguish between 4 cases: a) q < 1 In this case c n S n alost surely, if and only if li c n = 0, and for all the sequences ( j ) of the class R c j+1 k= j+1 β k θ k alost surely Thus, in the case of weak dependence the strong law of large nubers is the sae as in independent case b) q = 1 In this case c n S n alost surely, if and only if li nc n = = 0, and for all the sequences ( j ) of the class R c j+1 k= j+1 ( j+1 k + 1)β k θ k 0, alost surely c) q = 1 In this case c n S n alost surely, if and only if li c n = = 0, and for all the sequences ( j ) of the class R c j+1 β k θ k alost surely, j+1 k j+1: ( j+1 k) odd where the su is taken over all j + 1 k j+1, such that ( j+1 k) is an odd integer d) q > 1 In this case c n S n alost surely, if and only if li c nq n = = 0, and for all the sequences ( j ) of the class R c j+1 k= j+1 (1 q j+1 k+1 )β k θ k 0, alost surely

13 Scalar-nored sus of eleents of regressive sequences 377 Thus, these four cases illustrate how the rate of the dependence between the eleents of autoregressive sequence (η k ) influences on the for of the strong law of large nubers and the noralizing sequence (c k ) For the Gaussian Markov sequence (η k ) the result of Theore 21 specializes as follows Corollary 22 Let (θ k ) be a standard Gaussian sequence, ie (η k ) be a Gaussian Markov sequence Then c n S n alost surely, if and only if the following two conditions hold: 1) li c na(n, k) = 0, for any k 1; 2) for all the sequences ( j ) of the class R and any ε > 0 j=1 { exp ε c 2 j+1 k= j+1 (A( j+1, k)) 2 } < References [1] Buldygin, VV, The strong law of large nubers and convergence to zero of Gaussian sequences, Teor Veroyatn i Mat Stat, 19 (1978), (Russian); English translation in Theory Probab Math Statist, 19, [2] Buldygin, VV, Convergence of Rando Eleents in Topological Spaces, Kiev: Naukova Duka, 1980, (Russian) [3] Buldygin, VV and MK Runovska, On the convergence of series of autoregressive sequences, Theory of Stochastic Processes, 15(31), no 1, (2009), 7 14 [4] Buldygin, VV and MK Runovska, On the convergence of series of autoregressive sequences in Banach spaces, Theory of Stochastic Processes, 16(32), no 1, (2010), [5] Buldygin, VV and MK Runovska, Alost sure convergence of the series of Gaussian Markov sequences, Counication in Statistics - Theory and Methods, 40, no 19-20, (2011), [6] Buldygin, VV and SA Solntsev, Functional Methods in Probles of the Suation of Rando Variables, Kiev: Naukova Duka, 1989 (Russian)

14 378 MK Runovska [7] Buldygin, VV and SA Solntsev, Asyptotic Behavior of Linearly Transfored Sus of Rando Variables, Kluwer Acadeic Publishers, Dordrecht, 1997 [8] Gaposhkin, VF, On the suation alost everywhere of stationary sequences, Teor Veroyatn i Prienen, 32(1) (1987), (Russian); English translation in Theory Probab Appl, 32, [9] Gaposhkin, VF, On the dependence of the convergence rate in the strong law of large nubers for stationary processes on the rate of decay of the correlation function, Teor Veroyatn i Prienen, 26(4) (1981), , (Russian); English translation in Theory Probab Appl, 26, [10] Gaposhkin, VF, The law of large nubers for linear autoregression processes, Izv Vyssh Uchebn Zaved Mat, 4 (1988), (Russian) [11] Khintchine, A, Sur la loi des grands obres, CR Acad Sci Paris, 188(7) (1929), [12] Kologoroff, A, Sur la loi forte des grands nobres, CR Acad Sci Paris, 191(20), [13] Levy, P, Calcul des Probabilites, Gautier-Villars: Paris, 1925 [14] Loeve, M, Probability Theory, Springer Verlag, Berlin, 1977, Vol1, 4th ed; 1978, Vol2, 4th ed [15] Martikainen, AI, On necessary and sufficient conditions for the strong law of large nubers, Teor Veroyatn i Prienen, 24(4) (1979), , (Russian); English translation in Theory Probab Appl, 24, [16] Martikainen, AI and VV Petrov, On necessary and sufficient conditions for the law of the iterated logarith, Teor Veroyatn i Prienen, 22(1) (1977), 18 26, (Russian); English translation in Theory Probab Appl, 22, [17] Prokhorov, YuV, On the strong law of large nubers, Izv Akad Nauk SSSR Mat, 14(6) (1950), , (Russian) [18] Runovska, MK, Convergence of series of Gaussian Markov sequences, Teoriya Iovirnostei ta Mateatichna Statistika, 83 (2010), , (Ukrainian); English translation in Theory Probab and Math Stat, 83 (2011), [19] Runovska, MK, Convergence of series whose ters are eleents of ultidiensional Gaussian Markov sequences, Teoriya Iovirnostei ta Mateatichna Statistika, 84 (2011), , (Ukrainian); English translation in Theory Probab and Math Stat, 84 (2012), [20] Verbitskaya, IN, On conditions for the strong law of large nubers to be applicable to second order stationary processes, Teor Veroyatn i Prienen, 9(2) (1964), (Russian); English translation in Theory Probab Appl, 9,

15 Scalar-nored sus of eleents of regressive sequences 379 [21] Verbitskaya, IN, On conditions for the applicability of the strong law of large nubers to wide sense stationary processes, Teor Veroyatn i Prienen, 11(4) (1966), (Russian); English translation in Theory Probab Appl, 11, [22] Volodin, NA and SV Nagaev, A reark to the strong law of large nubers, Teor Veroyatn i Prienen, 22(4) (1977), (Russian); English translation in Theory Probab Appl, 22, MK Runovska Departent of Matheatical Analysis and Theory of Probability National Technical University of Ukraine (KPI) Kiev Polytechnic Institute Pereohy Ave, 37 Kiev, Ukraine atan@ntu-kpikievua