Central limit theorem for functions of weakly dependent variables
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1 Int. Statistical Inst.: Poc. 58th Wold Statistical Congess, 2011, Dublin (Session CPS058 p.5362 Cental liit theoe fo functions of weakly dependent vaiables Jensen, Jens Ledet Aahus Univesity, Depatent of Matheatical Sciences Ny Munkegade, 8000 Aahus C, Denak E-ail: Result We descibe a cental liit theoe fo the su S n = n i=1 X i, X i R p, whee the X i s can be appoxiated by weakly dependent vaiables. The poof is fo i Z, but can be diectly genealized to the case of a ando field, i Z d. The weak dependency is foulated though a set of σ-algebas, D j, j Z. The stong ixing coefficients of these ae α(k, l, d = sup P (A1 A 2 P (A 1 P (A 2, whee the supeu is taken ove sets A i σ ( D j : j I i, i = 1, 2, with I1 k, I 2 l, and the distance d(i 1, I 2 between the two sets I 1 and I 2 is at least d. The cental liit theoe has applications in the study of nonhoogeneous hidden Makov chains (Jensen Theoe 1. Assue that thee exist δ 0, ɛ 0 > 0, δ 1, δ 2 0, θ > δ 1 + δ 2 + ax { (2 + δ 0 /δ 0, 1 + δ 2, 2 } and constants c 0, c 1, c 2 such that (1 (2 (3 EX i = 0, E X i 2+δ 0 c 0, Va(a S n ɛ 0 n a a R p α(k, l, d c 1 k δ 1 l δ 2 ax{1, d} θ, N X j σ ( D k : d(k, j : E X j X j c 2 θ. Then we have that Va(S n 1/2 S n N p (0, I fo n. (Fo the case of a ando field, X i, i Z ν, the lowe bound on θ is ultiplied by ν. We divide the poof into a nube of subsections. In the fist two subsections we use tuncation to educe the poble to that of bounded vaiables. In the last section the ethod of Bolthausen (1982 is used fo the bounded vaiables. Tuncation We use the tunctation function T M whee T M (x equals x fo x M and equals Mx/ x othewise. Let Q M (x = x T M (x. Using that EX j = 0 we wite the su S n as S n = S n + S N, with (4 S n = [ TM (X j E(T M (X j ] and S n = [ QM (X j E(Q M (X j ], and the idea is to pove that a CLT fo S n fo any fixed M iplies a CLT fo S n. In the lea below we conside fixed values of j, k and fixed unit vectos a j and a k. We define U = a j X j, U M = a j T M (X j and ÛM = a j Q M (X j, and define V, V M and ˆV M siilaly with j eplaced by k. Lea 2. Thee exists a constant c 3, depending on c 1, c 2, δ 0 and θ only, such that [ 1/(2+δ Cov(U, V c 3 c c 2/(2+δ ] { } 0 κ, 0 ax 1, d(j, k whee κ = (θ δ 1 δ 2 δ 0 /(2 + δ 0 > 1. Poof. Following Deo (1973 we expand Cov(U, V = Cov ( U M + ÛM, V M + ˆV M into fou tes and bound each of these. Thus, using the siple bound Q M (x ( x /M 1+δ 0 x we find Cov ( U M, ˆV M 2ME ˆVM 2Mc 0 /M 1+δ 0 = 2c 0 /M δ 0.
2 Int. Statistical Inst.: Poc. 58th Wold Statistical Congess, 2011, Dublin (Session CPS058 p.5363 Siilaly, Cov (ÛM, V M 2c0 /M δ 0, and Cov (ÛM, ˆV M { Va( Û M Va( ˆV M } 1/2 c0 /M δ 0, since EÛ M 2 E X j 2+δ 0 /M δ 0. To bound Cov ( ( U M, V M we define U M = a j T M X j and V M = a k T M (Xk. Since T M (X j T M (Xj X j Xj we find Cov ( U M UM, V M 2ME U M UM 2c 2 M θ, Cov ( UM, V M VM 2c 2 M θ, and theefoe Cov(U M, V M Cov(UM, V M + 4c 2 M θ. Finally, we use the classical bound (Ibagiov and Linnik, 1971 [17.2.1] Cov(UM, V M 4M 2 α ( 2 + 1, 2 + 1, ax{0, d(j, k 2}. Putting all the tes togethe we obtain Cov(U, V 5c0 /M δ 0 + 4c 2 M θ + 4c 1 M 2 (2 + 1 δ 1+δ 2 ax { 1, d(j, k 2 } θ. Choosing = d(j, k/3 and M = c 1/(2+δ 0 0 d(j, k κ/δ 0 we get the esult of the lea. Lea 3. Let S n be defined in (4. Thee exists a function b(m with b(m 0 fo M such that fo all unit vectos a and fo all n we have Va ( a S n /n b(m. Poof. We use Lea 2 with the ando vaiable X eplaced by Z = Q M (X E(Q M (X. Let 0 < ξ < δ 0 be so lage that κ 1 = (θ δ 1 δ 2 ξ/(2 + ξ > 1. Reebeing the siple inequality Q M (x ( x /M α x we have ET M (X i = EQ M (X i c 0 /M 1+δ 0 and E Q M (X i 2+ξ c 0 /M δ 0 ξ. Thus, eplacing δ 0 by ξ we use the bound (5 E Zi 2+ξ 2 1+ξ { E QM (X i 2+ξ + EQM (X i 2+ξ } 2 1+ξ{( c 0 /M δ 0 ξ + ( c 0 /M 1+δ 0 2+ξ} = c0 (M, ξ. Futheoe, we can appoxiate Z by Q M (X E(Q M (X with a ean eo E Q M (X Q M (X E X X + E T M (X T M (X 2E X X 2c 2 θ. We can now use Lea 2 with δ 0 eplaced by ξ, c 0 eplaced by c 0 (M, ξ and c 2 eplaced by 2c 2. Fo soe constant c 3 and any unit vecto a we then have Cov ( a Q M (X j, a Q M (X k c 3 b(m ax { 1, d(j, k } κ1, whee b(m = c 0 (M, ξ 1/(2+ξ + c 0 (M, ξ 2/(2+ξ. Witing Va(S N as a double su of covaiances we find the esult of the lea with b(m = c 3 b(m[3 + 2/(κ1 1]. Fo (5 we see that c 0 (M, ξ tends to zeo, and theefoe b(m tends to zeo as M tends to infinity. Vaiance Witing Va(a S n as a double su of covaiances we get diectly fo Lea 2 the bound Va(a S n [ 1/(2+δ c 4 n with c 4 = c 3 c c 2/(2+δ ] 0 0 [3 + 2/(κ 1]. Fo Lea 3 we find Va ( a S n / n Va ( a S n/ n 2 Cov ( a S n / n, a S n/ n + Va ( a S n/ n 2 c 4 b(m + b(m 0 fo M.
3 Int. Statistical Inst.: Poc. 58th Wold Statistical Congess, 2011, Dublin (Session CPS058 p.5364 The assuption of a lowe bound on the vaiance of S n / n in Theoe 1 theefoe gives that (6 Va ( a S n / n /Va ( a S n/ n 1 fo M. As in Ibagiov and Linnik (1971, page 346 we have fo Lea 3 and (6 that it suffices to pove a CLT fo S n fo fixed M to obtain a CLT fo S n. Poof of cental liit theoe fo S n Let a be a fixed unit vecto and define Y i = a (T M (X i ET M (X i and S n = n i=1 Y i = a S n. We saw in the poof of Lea 3 that ET M (X i c 0 /M 1+δ 0 so that fo M sufficiently lage we have Y i M + 1. Futheoe, if we set Yi = T M (Xi ET M (X i we have E Y i Yi E X i Xi c 2 θ and Y i M + 1. We will use the ethod of poof fo Bolthausen (1982. Fo this we need the following estiates. Lea 4. Thee exists a constant c 4 such that Cov ( Y j, Y k c4 (M+1 2 ax{1, d(j, k} γ, Cov ( Y j Y k, Y Y s c4 (M+1 4 ax { 1, d({j, k}, {, s} } γ, and Cov ( Y j, Y k Y Y s c4 (M ax { 1, d(j, {k,, s } } γ, whee γ = θ δ 1 δ 2. Poof. The poof is based on successively eplacing Y i by Yi in the ean of a poduct of Y s. Thus, using the second inequality as an exaple, E(Y j Y k Y Y s E(Yj Yk Y Ys 4(M c 2 θ. Fo inequalities of this fo we obtain Cov(Y j Y k, Y Y s Cov(Yj Yk, Y Y s 2 4(M c 2 θ. Next, the stong ixing iplies (Ibagiov and Linnik, 1971 [17.2.1] Cov(Y j Yk, Y Y s 4(M c 1 [2(2 + 1] δ 1+δ 2 ax { 1, d({j, k}, {, s } 2} θ. Cobining the two inequalities and taking = d({j, k}, {, s}/4 we obtain the second inequality of the lea. Fo a nube we intoduce the notation S i,n =,d(i,j Y j and α n = E(Y i S i,n, i=1 whee eventually will be tending to infinity with n. Fo Lea 4 we find that Va ( Sn / n α n n = 1 n i=1,d(i,j> Cov(Y i, Y j 4c 4(M (γ 1 γ 1, whee γ 1 = θ δ 1 δ 2 1 > 0. Now, conside the case = n ω, with ω > 0. Then the ight hand side above tends to zeo. Since also we have fo (6 that Va( S n is of the sae ode as Va(S n, and we have assued the lowe bound ɛ 0 n fo the latte, we find that α n has a siila lowe bound and Va( S n /α n 1. We ust theefoe show that S n = S n / α n conveges to a standad noal distibution. Poof of CLT fo S n We follow the poof of Bolthausen (1982, whee the definitions of A 1, A 2 and A 3 below can be found. These tes depend on the aguent t of the chaacteistic function fo
4 Int. Statistical Inst.: Poc. 58th Wold Statistical Congess, 2011, Dublin (Session CPS058 p.5365 S n. One needs to show that E ( A 1 A 2 A 3 0. Using the expession in Bolthausen (1982 and Lea 4 we have with J = {1 i, i, j, j n : d(i, j, d(i, j }, (7 E A 1 2 t 2 = 2 Cov(Y i Y j, Y i Y j J = t2 { 2 Cov(Y i Y j, Y i Y j + J,d(i,i 3 c 4t 2 (M { 2 n( J,d(i,i <3 } Cov(Y i Y j, Y i Y j (k 2 γ + n( ( j γ} k=3 = O ( t 2 (M /n = O ( t 2 (M /n 1 2ω, whee the last expession follows upon taking = n ω. Fo the A 2 te we have fo Bolthausen (1982 and Lea 4 (8 E A2 nt 2 (M + 1 α 3/2 n ax i j,k=1,d(i,j,d(i,k Cov(Y j, Y k c 4nt 2 (M { 2+1 3/2 ( j γ} = O ( t 2 (M /n 1 ω 2. Finally, we need to conside A 3 = 1/2 n i=1 Y i exp { it( S n S i,n }, whee S i,n = S i,n / α n. Consideing the exponential pat, and eplacing all the vaiables by the appoxiating vaiables Yj, we see that E exp { it( S n S i,n } exp { it( S n S i,n } Using this we obtain t E ( S n S i,n ( S n S i,n t nc 2 θ. ( Cov Yi, exp{it( S n S i,n } Cov ( Yi, exp{it( S n S i,n} c2 θ + (M + 1 t nc 2 θ. Fo the ixing we have (Ibagiov and Linnik, 1971 [17.2.1] Cov ( Y i, exp{it( S n S i,n } 4(M + 1c 1 (2 + 1 δ 1 n δ 2 ( 2 θ. Taking = /3 and cobining the two bounds we find fo soe constant c 5 Cov ( Y i, exp{it( S n S i,n } { c 5 (M + 1n δ 2 θ+δ 1 + θ [1 + (M + 1 t n/ α n ] }. Since EY i = 0 the tes in A 3 ae covaiances, and taking = n ω we theefoe have the bound (9 EA 3 = O ( (M + 1[n ωθ+1 + n ω(θ δ 1+δ ]. Thus, if we choose ω such that ω < 1 2, ωθ 1 > 0 and ω(θ δ 1 δ > 0, which is possible fo the assuption on θ in Theoe 1, we see that all of (7, (8, and (9 tend to zeo. REFERENCES Bolthausen, E. (1982: On the cental liit theoe fo stationay ixing ando vaiables. Ann. Pobab., 10, Deo, C. M. (1973: A note on epiical pocesses of stong-ixing sequences. Ann. Pobab., 1,
5 Int. Statistical Inst.: Poc. 58th Wold Statistical Congess, 2011, Dublin (Session CPS058 p.5366 Ibagiov, I. A. and Linnik, Yu. V. (1971: Independent and Stationay Sequences of Rando Vaiables. Woltes-Noodhoff Publishing, Goningen. Jensen, J.L. (2005: Context dependent DNA evolutionay odels. Reseach Repot, No. 458, Depatent of Matheatical Sciences, Univesity of Aahus.
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