Strictly stationary solutions of spatial ARMA equations

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1 Strictly stationary solutions of spatial ARMA equations Martin Drapatz October 16, 13 Abstract The generalization of the ARMA time series model to the multidimensional index set Z d, d, is called spatial ARMA model. The purpose of the following is to specify necessary conditions and sufficient conditions for the existence of strictly stationary solutions of the ARMA equations when the driving noise is i.i.d. Two different classes of strictly stationary solutions are studied, solutions of causal and non-causal models. For the special case of a first order model on Z conditions are obtained, which are simultaneously necessary and sufficient. Keywords: causality, random fields, spatial ARMA model, strict stationarity. 1 Introduction Let d N, usually d > 1, and Y t t=t1,...,t d Z a d-dimensional complex-valued random d field living on a probability space Ω, F, P. If Y t t Z d fulfills the equations Y t φ n Y t n = Z t + θ n Z t n, t Z d, 1.1 n R n S where φ n n R, θ n n S C, R and S are finite subsets of N d \{} or more generally of Z d \{}, and Z t t Z d is an i.i.d. complex-valued random field on Ω, F, P, we call Y t t Z d an ARMA random field, where ARMA is short for autoregressive moving average. The spatial ARMA model defined by 1.1 is a natural generalization of the well-known ARMA time series model see e.g. Brockwell and Davis [4], Chapter 3 to higher dimensional index sets Z d, d > 1. The spatial ARMA model was considered long ago by Whittle [] and many others e.g. [19], [], [1] had been working on this topic. However, most work has been spent on weakly stationary solutions of the spatial ARMA model and their statistics. For the time series model d = 1, Brockwell and Lindner [5] obtained necessary and sufficient conditions for the existence of strictly stationary solutions of 1.1. In this article we generalize those results and obtain some necessary and some sufficient conditions for Institut für Mathematische Stochastik, TU Braunschweig, Pockelsstraße 14, D-3816 Braunschweig, Germany m.drapatz@tu-bs.de 1

2 the existence of strictly stationary solutions of 1.1, in terms of some moment conditions on the white noise Z t t Z d and zero sets of the characteristic polynomials Φz = 1 n R φ n z n, and Θz = 1 + n S θ n z n, z = z 1,..., z d C d, corresponding to the recurrence equation 1.1. The polynomial Φ is called autoregressive polynomial and Θ moving average polynomial we speak of polynomials even if R, S Z d. It is known that a sufficient condition for the existence of a weakly stationary solution, when usually Z t t Z d is considered to be only uncorrelated white noise with mean zero, is given by see Rosenblatt [14], page 6 Θe it Φe it dλ d t <, 1. T d where T d is the d-fold cartesian product of the factor space T = R/πZ, which we identify by π, π], and λ d is the Lebesgue measure on R d limited to T d. By spectral density arguments it can easily be shown that this condition is also necessary. Condition 1. will also play a decisive role, when strictly stationary solutions are considered. There are several differences between d = 1 and higher dimensional models with d > 1, which bring some difficulties: first of all, polynomials can not be factored completely like in one dimension, which implies that a quotient of polynomials in several variables may have common zeros that cannot be canceled out. Another difference is that even though Φe i may have zeros on T d, it is possible that 1. holds, even if Θz 1. Furthermore, we have to deal with multiple sums k N d X k for some random field X k k Z d, which do not necessarily converge absolutely. Therefore a type of convergence defined by Klesov [1], namely almost sure convergence in the rectangular sense, will be used. The article is structured as follows: in Section we study linear strictly stationary solutions. After that in Section 3 we go on considering strictly stationary causal solutions, without assuming them a priori to be linear. Then in Section 4 a full characterization of necessary and sufficient conditions for the existence of strictly stationary causal solutions of a first order autoregressive model in dimension two will be given. The following notation will be used: vector-valued variables will be printed bold and the multi-index notation z n = z n 1 1 z n d d, z = z 1,..., z d C d, n = n 1,..., n d Z d, e it = e it 1,..., e it d, t T d, will be applied. To indicate that two random variables X and Y are independent, the symbol X Y will be used. Furthermore the Backward Shift Operator B = B 1,..., B d, where B i shifts the ith coordinate back by one, i.e. for the ith unit vector e i in R d we have B i Z t = Z t ei, i = 1,..., d, t Z d, will be used to write the ARMA equation 1.1 in a compact form as ΦBY t = ΘBZ t, t Z d.

3 The Hilbert space of functions f : T d C, which are square integrable with respect to λ d, will be denoted by L T d. If condition 1. is fulfilled, the existence of a Fourier expansion Θe it Φe it = k Z d ψ k e ikt, ψ k k Z d C, t T d, 1.3 where kt = k t = d i=1 k it i denotes the Euclidean inner product on R d, is assured, see Shapiro [16], Theorem.. Plugging 1.3 into 1., it is easy to see that the coefficients ψ k k Z d are square summable, i.e. k Z ψ d k <. By H we denote the Banach space containing all functions f : D d C holomorphic on the open unit polydisc D d = {z = z 1,..., z d C d : z i < 1, i = 1,..., d} and fulfilling f 1 H := sup r<1 π d T d fre it dλ d t <. If a function f : D d C is holomorphic, it admits a power series expansion fz = k N a d k z k, see Range [1], Theorem 1.6. Thus, a function f : D d C H admits a representation fz = k N a d k z k and f H, if and only if k N a d k <. To see that, notice that f H = sup a k r k. r<1 k N d Hence, each function f H can be identified with its boundary function g : T d C, whose Fourier expansion is given by ge it = k N a d k e ikt. Further f H = sup k r<1 k N a r k = a k = 1 ge it dλ d t =: g π d L T d, d k N d T d so that H can be identified with a closed subspace of L T d, more precisely, the space of all functions g L T d, whose Fourier coefficients a k k Z d vanish for k Z d \N d. The space H is called Hardy space. For more details about Fourier Analysis and Hardy spaces in several variables see Shapiro [16] or Rudin [15]. Beside Fourier expansions, Laurent expansions in several variables will be utilized. All results from function theory in several variables used in this work can be found in Range [1]. Linear Strictly Stationary Solutions In this section we introduce the notion of linear strictly stationary ARMA random fields and establish necessary and sufficient conditions for the existence of solutions of the ARMA equations for this class of random fields. In the whole section we assume that R and S are subsets of Z d \{}. Definition.1. A random field Y t t Z d, which solves the ARMA equation 1.1 where Z t t Z d is an i.i.d. noise, is called linear strictly stationary solution, if there are coefficients ψ k k Z d C, such that Y t = k Z d ψ k Z t k, t Z d, where the right-hand side converges almost surely absolutely. 3

4 Obviously, a linear strictly stationary solution is indeed strictly stationary. Theorem.. Let R and S be subsets of Z d \{} and Z t t Z d an i.i.d. random field. The ARMA equation 1.1 admits a linear strictly stationary solution if and only if Θe i Φe i L T d, and if Y t = k Z d ψ k Z t k, t Z d,.1 converges almost surely absolutely, where Θe it Φe it = k Z d ψ k e ikt, t T d,. denotes the Fourier expansion of Θe i /Φe i. If these two conditions are satisfied, then a linear strictly stationary solution is given by.1. Proof. Suppose both conditions are fulfilled. Applying the operator ΦB on Y t t Z d defined in.1 yields as ΦBY t =Y t n R φ n Y t n = ψ k φ n ψ k n Z t k. k Z d n R } {{ } =:ξ k The random field Y t t Z d solves the ARMA equations, if the coefficients ξ k k Z d fulfill ξ k := ψ k θ k, k S\{}, φ n ψ k n = 1, k =,.3 n R, otherwise. To prove the validity of these equalities we compare the coefficients ξ k k Z d with those of the corresponding Fourier series. Multiplying both sides of equation. by Φe it yields Θe it = 1 + θ n e int = Φe it ψ k e ikt = k Zd ψ k φ n ψ k n e ikt. n S k Z d n R.4 Comparing the coefficients in equation.4, the validity of.3 is obtained, which completes the proof of sufficiency. 4

5 Suppose the random field Y t t Z d is a linear strictly stationary solution of the ARMA equations. Thus, it has a representation Y t = k Z d ψ k Z t k for t Z d for some sequence ψ k k Z d C, where the right-hand side converges almost surely absolutely. By an application of Theorem of Chow and Teicher [6] this implies the square summability of the coefficients ψ k k Z d. The Theorem of Riesz-Fischer see Stein and Weiss [17], Theorem 1.7 now implies that there exists a function Ψe i in L T d, whose Fourier coefficients are precisely ψ k k Z d. Hence the Fourier expansion of Ψe i is given by Ψe it = k Z d ψ k e ikt for t T d. Yet again, we can compare the coefficients of the ARMA equation ΦBY t = ΘBZ t and those of the product Φe i Ψe i and conclude Φe it Ψe it = Θe it, t T d..5 We define the measurable set N := {t T d : Φe it = } and obtain by equation.5 Θe it Φe it dλ d t = Ψe it dλ d t <. T d \N T d \N Furthermore, by Theorem 3.7 of Range [1], the set N is a λ d nullset. Thus, Θe i /Φe i L T d and because of the uniqueness of the Fourier expansion, the Fourier coefficients of Θe i /Φe i are given by ψ k k Z d. An immediate question is under which conditions the right-hand side of equation.1 converges almost surely absolutely. Before giving a sufficient condition in Proposition.4, we need the following lemma. Lemma.3. For n, d N denote the cardinality of the set { k Z d : k k d = n } by h d n. Then h d n can be estimated from above by C d n d 1 for some constant C d >. Proof. For d = 1 we have h 1 n = {k Z : k = n} =. Suppose the assumption is valid for d N. Then the following identity holds for d + 1 { k Z d+1 : k k d+1 = n } n = {k Z d+1 : k= d k i = n k, k d+1 = k}. i=1 Each set of this union with k n has cardinality less than or equal to C d n k d 1 by assumption and for k = n the cardinality is two. Thus we can conclude h d+1 n C d n d + C d+1 n d for some constant C d+1 >. We define for z the postive part of the natural logarithm as log + z := maxlogz,. In the following proposition sufficient conditions for the existence of a linear strictly stationary solution are given. Proposition.4. Let R and S be subsets of Z d \{}. If the autoregressive polynomial Φe i has no zero on T d, then for appropriate r = r 1,..., r d, ρ = ρ 1,..., ρ d, r i < 1 < ρ i, i = 1..., d, a Laurent expansion of Θe i /Φe i exists given by Θz Φz = k Z d ψ k z k, z Kr, ρ := {z = z 1,..., z d C d : r i < z i < ρ i, i = 1,..., d}. 5

6 If further then E log d + Z 1 <, Y t := k Z d ψ k Z t k, t Z d, converges almost surely absolutely. In particular, the random field Y t t Z d solves the ARMA equation 1.1. Proof. If the autoregressive polynomial Φe i does not possess any zeros on T d, then the quotient Θz/Φz is holomorphic in Kr, ρ, where r i < 1 < ρ i for suitable r = r 1,..., r d, ρ = ρ 1,..., ρ d. Furthermore Proposition 1.4 in Range [1] assures the existence of a Laurent expansion Θz/Φz = k Z d ψ k z k, z Kr, ρ, r < 1 < ρ, and the validity of the Cauchy estimates in d variables, i.e. there are constants M, c > such that ψ k Me c k k d, k Z d. For n N the number of possibilities of k Z d fulfilling k k d = n can be estimated from above by C d n d 1, C d >, see Lemma.3. Thus, for c, c it follows P ψ k Z t k > e c k kd PM Z t k > e c c k kd k Z d k Z d = k Z d Plog + M Z > c c k k d 1 + C d n=1 n d 1 Plog + M Z > nc c. We define the random variable X = log + M Z /c c. Using the two inequalities PX > n PX > x for x n 1, n], n N, n x for x n 1, n], n N\{1}, the last series in equation.6 can be estimated from above as follows n d 1 PX > n n= n= n n 1 PX > x x d 1 dx = d 1 PX > x x d 1 dx d 1 d 1.6 EX d = d 1 d E log+ M Z d <,.7 c c 6

7 where the last inequality is valid, because the expected value in.7 is finite, if and only if E log d + Z <. Applying the Borel Cantelli Lemma implies that the event { ψ k Z t k > e c k k d for infinitely many k Z d } has probability zero. The almost sure majorant k Z e d c k k d is absolutely convergent, and hence the series Y t = k Z ψ d k Z t k converges almost surely absolutely for all t Z d. For d = 1 the condition Φz for all z C, z = 1, is a necessary and sufficient condition for the uniqueness of a strictly stationary solution, provided one exists, see Brockwell and Lindner [5]. For d > 1 we do not know whether the analog condition Φe it for all t T d is sufficient for the uniqueness of linear strictly stationary solutions. However, the necessity of this condition is shown in the following lemma. Lemma.5. Suppose Y t t Z d is a strictly stationary solution of 1.1. Suppose further that Φe i has a zero λ T d. Finally, suppose the underlying probability space is rich enough to support a random variable U, which is independent of Y t t Z d and uniformly distributed on [, 1]. Then Y t + e iπu e itλ t Z d is another strictly stationary solution of 1.1. In particular, the strictly stationary solution of 1.1 is not unique. Proof. It is easy to see that the random field W t = e iπu e itλ, t Z d, is strictly stationary. Because of the independence of U and Y t t Z d, the random field X t t Z d defined by X t = Y t + W t, is also strictly stationary. Furthermore we have ΦBW t = W t n R φ n W t n = e itλ e iπu Φe iλ =, which shows that X t t Z d is another solution of the ARMA equation 1.1. Notice that, if the conditions of the preceding lemma are fulfilled and Y t t Z d is a linear strictly stationary solution, then the random field Y t +e iπu e itλ t Z d is an example for a strictly stationary solution which is not linear. 3 Causal solutions In this section we study necessary conditions and sufficient conditions for the existence of causal solutions. We define for t = t 1,..., t d, s = s 1,..., s d Z d the index sets {s t} induced by the relation on Z d : {s t} := {s Z d : s i t i, i = 1,..., d}. Definition 3.1. A strictly stationary random field Y t t Z d, which fulfills the ARMA equation 1.1, is called causal solution of the spatial ARMA model, if Y t is measurable with respect to σz s : s t for each t Z d. 7

8 When considering causal solutions, it makes sense to restrict the index sets R, S in equation 1.1 to subsets of N d \{}. We assume this throughout the whole section. We will see that the symmetrization Ỹt t Z d of a causal solution Y t t Z d admits a linear representation Ỹt = k N d ψ k Zt k for some coefficients ψ k k N d C and a symmetrization Z t t Z d of Z t t Z d. But in contrast to the definition of linear strictly stationary solutions, a specific type of convergence is not required by Definition 3.1. However, implicitly this sum has to convergence almost surely in the rectangular sense, as we will see later on in Theorem 3.7. Definition 3. Klesov [1], Definitions 1 and 3. Let Z k k N d be a real-valued random field. The multiple series k N Z d k converges almost surely in the rectangular sense, if the limits lim N 1k k k 1 = for all sequences N 1k,..., N dk k N N d with minn 1k,..., N dk k almost surely exist and coincide. For this mode of convergence a generalization to multiple series of the three series theorem of Kolmogorov is valid. Precisely, the following theorem holds. Theorem 3.3 Klesov [11] and Klesov [1], Theorem C. Let X n n N d be a real-valued random field of independent random variables and define Xn c := X n 1 { Xn <c}. Then almost sure convergence of X n, 3.1 in the rectangular sense and the condition n N d N dk k d = Z k, P X nk > ɛ k, ɛ >, 3. for all sequences n k k N with maxn 1k,..., n dk, are equivalent to the convergence of the following three series for some c >, and hence for any c > : P X n c <, A n N d EXn, c n N d VarXn. c B C n N d Here, convergence of B is to be understood as almost surely rectangular. If d = 1 and n=1 X n converges almost surely, condition 3. is fulfilled automatically. For d > 1 the following example from Klesov [1] shows that this condition is not fulfilled in general. 8

9 Example 3.4. Define Xi, j = 1 j i for i 1 and j and for the rest Xi, j =. Then the convergence in the rectangular sense of i,j N Xi, j is clear, since n Xi, j = for all n. i,j=1 But the series A i,j= P Xi, j > c diverges for all c >. Notice further that 3. does not hold in this example. In addition Klesov [1] shows that the condition 3. can be dropped, if X n n N d is symmetric or the random variables are positive. We state it in the following corollary. For our purposes, mainly the symmetric case is relevant. Corollary 3.5 Klesov [1], Corollaries 3 and 4. Let X n n N d be a real-valued random field of independent random variables. Then almost sure convergence of 3.1 in the rectangular sense is equivalent to the convergence of A and C for some c >, and hence for any c >, if the random variables X n n N d are symmetric, and equivalent to the convergence of A and B for some c >, and hence for any c >, if the random variables X n n N d are positive. A random field Z t t Z d is called deterministic, if there is a constant K such that PZ t = K = 1 for all t Z d. Consider a nondeterministic real-valued i.i.d. stochastic process X n n N and coefficients ψ n n N R. If the series n N ψ nx n converges almost surely absolutely, then by application of Theorem of [6] it can be concluded that n N ψ n <. The same is true for multiple series, which converge almost surely in the rectangular sense. Theorem 3.6. Suppose X n n N d is a nondeterministic real-valued i.i.d. random field and ψ n n N d are real coefficients. Furthermore suppose the random variables X n n N d are symmetric or positive. If the multiple series n N d ψ n X n converges almost surely in the rectangular sense, then n N d ψ n <. Proof. Suppose n N d ψ n X n converges almost surely in the rectangular sense. Then Corollary 3.5 and the remark preceding it imply that P ψ nk X nk > ɛ ɛ >, k, for all sequences n k k N with maxn 1k,..., n dk. By assumption X n n N d is an i.i.d. and nondeterministic random field. This implies that some ɛ > exists such that P X n > ɛ = P X > ɛ >, and in particular X nk does not converge in probability to zero, if maxn 1k,..., n dk for k. Hence we can conclude ψ nk for all sequences n k k N with maxn 1k,..., n dk. Defining Y n := X n 1 { ψnxn <1} we can conclude by Theorem 3.3 n N d ψ nvary n < 3.3 9

10 if ψ n =, then EY n or VarY n need not to be defined, in which case we interpret VarY n as being infinity and ψ n VarY n to be equal to zero. Next, we claim that lim inf N min VarY n >. 3.4 n N If this were not true, there must be a subsequence n k k N N d such that maxn 1k,..., n dk and VarY nk as k. Thus, Y nk EY nk k in L P, and hence X 1 { ψnk X <1} EX 1 { ψnk X <1}, which is equal in distribution to Y nk EY nk, converges in probability to zero as k. But this implies that X is deterministic and we have a contradiction. Hence 3.4 is satisfied and together with 3.3 we obtain k N d ψ k <. With the aid of the preceding theorem, we are able to prove necessary conditions for the existence of causal solutions. Theorem 3.7. Assume that Z t t Z d is i.i.d. nondeterministic noise and R, S N d \{} and that the ARMA equation 1.1 admits a causal solution Y t t Z d. Then the following two conditions are satisfied: i Θz Φz H, z D d. ii Let Y t, Z t t Z d be an independent copy of Y t, Z t t Z d and define the symmetrizations { Ỹt, Z Y t, Z t, P Y and P Z symmetric, t := Y t Y t, Z t Z t, 3.5 otherwise. Let further Θz Φz = α k z k, z D d, 3.6 k N d denote the power series expansion of Θz/Φz, then Ỹt t Z d is a solution of the symmetrized ARMA equation ΦBỸt = ΘB Z t, t Z d, and given by Ỹ t = α k Zt k, t Z d, k N d where the convergence of the right-hand side is almost surely rectangular. In particular, if P Z is symmetric, then there is at most one symmetric causal solution. Proof. Assume Y t t Z d is a causal solution of the ARMA equation 1.1. Then the symmetrizations Ỹt t Z d and Z t t Z d fulfill the equation ΦBỸt = ΘB Z t, t Z d, and obviously Ỹt t Z d is a causal solution of this equation. Reorganizing this ARMA equation, we get Ỹ t = φ n Ỹ t n + Z t + θ k Zt k, t Z d. n R n S 1

11 Now we replace each Ỹt n on the right side by the ARMA equation. We do so as long as none of the random variables Ỹt s, s N 1,..., N d N d remains on the right-hand side of this equation. Defining for N = N 1,..., N d N d the index sets I N = {k N d : k i N i, i = 1,..., d}, B N = {k N d : k = m+n, m I N, n R S}\I N, we get for N = N 1,..., N d N d an equation like the following Ỹ t = α n,n Zt n + β n,n Zt n + γ n,n Ỹ t n n I N n B N n B N =: A t,n + B t,n + C t,n, N N d, 3.7 where α n,n n IN, β n,n n B S N,γ n,n n B R N are some complex coefficients and A t,n, B t,n and C t,n denote the first, second and third sum, respectively. Using the causality of Ỹt t Z d, notice that and furthermore we observe that A t,n B t,n, C t,n N N d, t Z d, α n,n = α n,n N > N, n I N. 3.8 Because of equation 3.8 we write from now on rather α n than α n,n. In the following we want to show that A t,n is not converging in probability to infinity as minn 1,..., N d using a technique adapted from [1]. The sum A t,n is not converging in probability to infinity, if we can find some constants K, ɛ > such that for every sequence N k k N, N k = N 1k,..., N dk N d, with minn 1k,..., N dk as k the following holds P A t,nk < K ɛ, k N. 3.9 By stationarity of Ỹt t Z d there are some constants K > and ɛ, 1 such that 4 P Ỹt < K 1 ɛ t Z d. Using equation 3.7 this implies for ɛ, 1 4 P A t,nk + B t,nk + C t,nk < K 1 ɛ, k N. Using the inequalities Rz, Iz z, z C, it follows P RA t,nk + B t,nk + C t,nk < K 1 ɛ, and P IA t,nk + B t,nk + C t,nk < K 1 ɛ, k N. 3.1 We will now show that the following holds for c 1, 1 ɛ P RA t,nk < K c, P IA t,nk < K c, k N Suppose 3.11 is not true. Then we can find k 1, k N such that one of the following inequalities is true P RA t,nk1 K 1 c, P IA t,nk K 1 c. 11

12 So by the symmetry of A t,n we have PRA t,nk1 K 1 c, PRA t,n k1 K 1 c, or 3.1a PIA t,nk K 1 c, PIA t,n k K 1 c, 3.1b and obviously PRB t,nk1 + C t,nk1 1 or PRB t,nk1 + C t,nk1 1. Suppose without losing the generaltity PRB t,nk1 + C t,nk1 1. By symmetry of RỸt = RA t,nk1 + B t,nk1 + C t,nk1 and independence of A t,nk1 and B t,nk1 + C t,nk1 it follows in case 3.1a P RA t,nk1 + RB t,nk1 + C t,nk1 K = PRA t,nk1 + RB t,nk1 + C t,nk1 K Similarly, in case 3.1b we obtain P IA t,nk + IB t,nk + C t,nk K 1 c PRA t,nk1 K, RB t,nk1 + C t,nk1 = PRA t,nk1 K PRB t,nk1 + C t,nk1 1 c > ɛ > ɛ The equations 3.13 and 3.14 provide a contradiction to equation 3.1. Hence equation 3.11 is fulfilled and equation 3.9 follows easily with possibly different constants ɛ, K. In particular we showed that n I Nk α n Zt n does not converge in probability to infinity as k. Thus, by Theorem 3.17 of Kallenberg [9] we can conclude that n I Nk α n Zt n converges almost surely. Furthermore we notice by equation 3.7 that B t,nk + C t,nk converges also almost surely as k, and it is measurable with respect to σ Z s, s t N l, l N. Hence we can deduce that the limit of B t,nk + C t,nk is measurable with respect to the tail σ-field σ Z s, s t N k. k N By Kolmogorov s zero-one law this σ-field is P-trivial. Hence the limit of B t,nk + C t,nk is almost surely constant, which we denote by u for the moment. Altogether we have Ỹ t = u + lim a.s, α n Zt n k n I Nk for every sequence N k k N, N k = N 1k,..., N dk N d, with minn 1k,..., N dk as k. By symmetry of Ỹt t Z d and Z t t Z d, we must have u = almost surely. Further, n N α d n Zt n converges almost surely in the rectangular sense. Applying Theorem 3.6 it follows that α n <. n N d 1

13 The proof is finished if we can show that Θz/Φz H expansion of this function is given by and the power series Θz Φz = α k z k, z D d k N d To do so, we apply the operator ΦB to both sides of equation 3.7 and get by 1.1 ΦBỸt = α n ΦB Z t n + β n,n ΦB Z t n + γ n,n ΘB Z t n n I N n B N n B N = ΘB Z t, N = N 1,..., N d N d. Because of the independence of Z t t Z d and the assumption that it is nondeterministic we can conclude that the coefficients of both sides of this equation are equal. Thus for each N = N 1,..., N d with I N S we have α = 1 and α k { θk, k S, φ n α k n =, k I N \S {}. n R,n k Hence we observe for all t T d 1 φ n e int α k e ikt = α k n R k N d k N d =1 + n S n R,n k φ n α k n e ikt θ n e int = Θe it The zero set of Φe i is a null set of the d-dimensional Lebesgue measure. Because of this and equation 3.16, we conclude Θe i /Φe i H L T d and that is equation In the case that P Z is symmetric, we can now give the following necessary and sufficient condition for the existence of symmetric causal solutions: Corollary 3.8. Assume Z t t Z d is i.i.d. nondeterministic symmetric noise and R, S N d \{}. Then the ARMA equation 1.1 admits a causal solution Y t t Z d if and only if Θz Φz H, z D d, and k N d α k Z t k converges almost surely in the rectangular sense, where α k k N d is given by 3.6. In that case, Y t := k N d α k Z t k defines the unique symmetric causal solution. Proof. Necessity has been shown in Theorem 3.7, and sufficiency follows as in the proof of Theorem.. 13

14 In the following theorem we give sufficient conditions for the existence of a causal solution. Notice that under condition i the convergence is even almost surely absolutely. Theorem 3.9. A causal solution of the ARMA equation exists, if one of the following conditions is fulfilled: i E log d + Z < and Φz z D d. ii E Z Θz <, EZ = and Φz H. In both cases the quotient Θz/Φz admits a power series expansion, given by and a causal solution is given by Θz Φz = ψ k z k, z = z 1,..., z d D d, k N d Y t = k N d ψ k Z t k, t Z d Proof. In case i by the condition that Φz has no zero on D d the existence of a multidimensional power series expansion is assured. The remaining proof is almost the same as the proof of Proposition.4. The assumptions in case ii assure L P-convergence of Furthermore it is easy to see that 3.17 solves the ARMA equations in both cases. Having derived necessary conditions and sufficient conditions, we want to discuss the crucial condition that the quotient of the ARMA polynomials lies in H. In the time series model d = 1, if Φz and Θz have no common zeros, a necessary and sufficient condition for the existence of strictly stationary solution is Φz for z = 1 and E log + Z < if Φ is not constant, see Brockwell and Lindner [5]. We will see that in contrast to the time series model for d > 1 it is not necessary that the analog condition Φe it for t T d holds, cf. the upcoming Example 3.1. A polynomial in two or more variables can in general not be factored as in one variable. If a polynomial px 1,..., x n admits a factorization p = qr, where q and r are nonconstant polynomials of n or less variables, then it is called reducible, otherwise irreducible. Every polynomial of several variables admits a factorization into irreducible factors, which is essentially, except for multiplication with constants, unique, cf. Bôcher [3], Chapter 16. If this factorization consists of only one nonconstant irreducible factor, then the polynomial is irreducible. The following result will be useful to exclude zeros of Φ on the closed unit disc if d = and Θ 1, cf. Corollary Theorem 3.1. Suppose Φ : C d C is an irreducible polynomial in d variables and further, if arbitrary d 1 variables are fixed, the polynomial in the remaining variable is not identically zero. If Φ has a root t = t 1,..., t d Dd \T d, i.e. Φt =, then Φ has also roots inside the open unit polydisc D d. 14

15 Proof. Suppose t = t 1,..., t d Dd \T d is a root of Φ. Then for at least one i {1,..., d} we have t i D, and for at least one i {1,..., d} we have t i T. Define { } { } I := i {1,..., d} : t i T, N := i {1,..., d} : t i D, I N = {1,..., d}. Without loss of generalization, we assume I = {1,..., j} and N = {j + 1,..., d}. We fix all variables except t j I and t d N and consider the two variable polynomial pt j, t d := Φt 1,..., t j 1, t j, t j+1,..., t d 1, t d = n a k t j t k d, n N, where the coefficients a k t j are themselves polynomials in one variable t j. We have pt j, t d =. Suppose a nt j. Then, by Theorem of [], the polynomial roots x i t, i = 1,..., n of the equation pt, x i t = can be chosen to be continuous in t in a neighborhood of t j. Thus, there exists t 1 j, t 1 d D such that pt 1 j, t 1 d =. Now suppose a n t j =. Then there exists 1 k < n such that a k t j or otherwise the polynomial pt j, is identically zero, which is excluded by the assumptions of the theorem. In the first case we can use the same argument as before. Applying this argument inductively for all i I yields the statement of the theorem. k= If we consider a polynomial Φ : C C in two variables, the assumption that Φ is irreducible implies the second condition in the preceding theorem: if Φz 1, z = n k= a kz 1 z k is identically zero, the first variable being fixed, then the coefficients a k, which are polynomials themselves, have a common zero and hence the polynomial can be factorized. Thus, by Theorem 3.1 for d = a root in D \T of an irreducible polynomial implies a root inside D. However, in three variables this is not true, consider e.g. Φz 1, z, z 3 = 1 z 1 z z z3. Fixing z 1, z = 1, 1 the polynomial Φ1, 1, z 3 is identical to zero, but Φ can not be factorized. Notice that for Φ 1 H it is necessary and sufficient that Φz on D d and 1 Φe it dλ d t < T d where the integral does not depend on the values on D d \T d. With the help of Theorem 3.1, we now establish the following necessary condition in the spatial autoregressive model for d = : Corollary Suppose Z t t Z is i.i.d. nondeterministic and R N \{}. A necessary condition for the existence of a causal solution in the autoregressive model 1.1, where Θz 1, z 1, is given by Φz 1, z z 1, z D. 15

16 Proof. By Theorem 3.7 we know that Φ 1 z 1, z H is a necessary and sufficient condition for the existence of a causal solution. This implies directly that Φz 1, z can not possess any root on D. Now assume Φz 1, z = n i=1 Φ iz 1, z is a factorization of Φ, where each Φ i z 1, z is irreducible. If one factor Φ i z 1, z = Φ i z 1 only depends on one variable, then it follows Φ i z 1 z 1 C : z 1 1, since, if z 1 < 1 is a zero, then Φz 1, z will have a zero on D. If z 1 C with z 1 = 1 is a root, then Φ 1 z 1, z can not be square integrable. If a factor Φ i depends on two variables, then we can apply Theorem 3.1 and it follows Φ i z 1, z on D \T. It remains to show that Φe i, e i on T. Suppose w T is a zero of Φe i, e i. Then by the mean value theorem, for an arbitrary norm on R and some C > But this implies Φe iw+h = Φe iw+h Φe iw C h, h T. T dh Φe iw+h C 1 T dh =, 3.19 h where the latter integral is infinite by simple calculus. This contradicts Φ 1 H. Altogether we showed that no zero on D can exist. We go on discussing the relation between zeros on T d and the finiteness of The following example from [14] shows that for d 3 it is possible to have roots on T d and still Φ 1 z H holds. Example 3.1. Consider for d = 5 the function Φz = z i, z = z 1,..., z 5 C 5. i=1 We will show Φ 1 e it L T 5, and this implies Φ 1 z H by noticing that the only root in D 5 is 1 = 1,..., 1. Utilizing the Taylor expansion e ih j = 1 ih j h j + Oh 3 j, h j, j = 1,..., 5, we estimate Φe ih 1 = 5 5 e ih j 1 j=1 = = ihj + h j + Oh 3 j j=1 5 ih j + h j 1 + j=1 5 j=1 Oh3 j j=1 ih j + h j. Observe that 5 j=1 Oh3 j j=1 ih j + h j as max h 1,..., h 5. 16

17 Hence, there are ɛ > and C 1, C > such that dh Φe ih C dh 1 + C Φe ih 1 + C T 5 h <ɛ where denotes the Euclidean norm. C 1 + 5C h <ɛ h <ɛ dh j=1 ih j + h j dh h 4 <, Rosenblatt [13], p. 8, states that the reciprocal of the similar polynomial Φz 1, z, z 3 = z 1 + z + z 3 for d = 3 is also in H. While for d = for autoregressive models Φe it for all t T is necessary, this is no longer the case for ARMA models, as the following example shows. Example Consider the two-dimensional ARMA model Y t 1 B 1Y t 1 B Y t = Z t B 1 Z t B Z t + B 1 B Z t, t = t 1, t Z. The corresponding moving average and autoregressive polynomials are given by Θz 1, z = 1 z 1 1 z, Φz 1, z = 1 1 z 1 1 z, z 1, z C. Notice that both polynomials have a common zero z 1, z = 1, 1 on T. We define H as usually as the vector space of all holomorphic functions f : D d C, which are bounded on D d. Then Θz 1, z /Φz 1, z H H, as can be seen by following estimation: Θz 1, z Φz 1, z = 1 z 1 1 z 1 z z 1 = z z [R 1 1 z z 1 ] 4, z 1, z D. In difference to the one-dimensional case the common root of the nominator and denominator can not be canceled out, because the polynomials do not factorize. In some sense, the zero of the nominator covers for the zero of the denominator, resulting in the square integrability. 4 The spatial autoregressive model of first order In the foregoing section we were able to specify necessary conditions and sufficient conditions for the existence of causal solutions, in terms of the zero set of the ARMA polynomials. However, we could not give necessary moment conditions on the noise Z t t Z d. It turns out that in difference to the one-dimensional case, where the asymptotics of the coefficients of the Laurent expansion Θz Φz = ψ k z k, k Z 17

18 can be easily completely determined in dependence of the zeros of Φz, for d > 1 it is difficult to determine the asymptotics of the corresponding Laurent or power series expansion. To specify necessary moment conditions, lower bounds on the decay of the coefficients are needed. However, even though in general it seems difficult to determine lower bounds or the exact asymptotics, for a specific model we are able to determine necessary moment conditions. In this section we want to establish a full characterization of necessary and sufficient conditions for the existence of causal solutions of the autoregressive model of first order with real coefficients in dimension two: Consider the spatial autoregressive model defined by the equations Y t1,t φ 1 Y t1 1,t φ Y t1,t 1 φ 3 Y t1 1,t 1 = Z t1,t, t 1, t Z, 4.1 where φ 1, φ, φ 3 R, φ 1, φ, φ 3,, and Z t t Z is an i.i.d. complex-valued random field. We want to establish necessary and sufficient conditions for the existence of a causal solution. To do so, several auxiliary results are needed. First, we want to determine the coefficients, which solve the to the model 4.1 corresponding partial difference equation ψ n,k = φ 1 ψ n 1,k + φ ψ n,k 1 + φ 3 ψ n 1,k 1, n, k N, n, k,, 4. ψ n, = φ n 1, ψ,k = φ k for n, k N, 4.3 where 4.3 are the boundary conditions and convention ψ n,k = for n, k Z \N is used. It is associated with 4.1 by the equation ψ n,k z1 n z k 1 φ 1 z 1 φ z φ 3 z 1 z = 1. n,k= The solution of this partial difference equation is determined in [7]: Lemma 4.1. The unique solution of 4. with boundary conditions 4.3 is given by n ψ φ 1,φ,φ 3 k n + k j n,k := ψ n,k = φ n j 1 φ k j φ j = j= n j= j n j k j k φ n j 1 φ k j φ 1 φ + φ 3 j, n, k N. 4.5 The formulas 4.4 and 4.5 are also valid if some of the coefficients φ 1, φ, φ 3 are equal to zero. The numbers ψ φ 1,φ,φ 3 n,k are called weighted Delannoy numbers. They can be interpreted as the number of weighted paths from, to n, k in the two-dimensional lattice N, when only moving with steps 1,,, 1 and 1, 1 is allowed and related weights φ 1, φ and φ 3 respectively. To establish later on necessary moment conditions on the noise Z t t Z, the asymptotics of ψ φ 1,φ,φ 3 n,k for n, k have to be known. Hetyei [8] discovered that the weighted Delannoy numbers are related to Jacobi polynomials. For n N the n-th Jacobi polynomial P α,β n x of type α, β, α, β > 1, is defined as P n α,β x = n n! 1 1 x α β dn 1 + x 1 x n+α 1 + x n+β, x 1, 1. dx n 18

19 The Jacobi polynomial P n α,β x is indeed a polynomial of degree n on 1, 1 and can therefore be extended to x R. The following relationship is valid, see Theorem.8 of [8]: Theorem 4.. For β N and φ 3 we have ψ φ 1,φ,φ 3 k+β,k = φ β 1 φ 3 k P,β k φ 1φ φ 3 1, k N. Therefore the asymptotic behaviour of the coefficients depends on the asymptotics of the Jacobi polynomials, which have been studied extensively. Wong and Zhao [, Theorem 5.1] established an asymptotic expansion for Jacobi polynomials with explicit error term. Accordingly, the asymptotic expansion of order p N can for N := n + β+1 with β N be written as cos θ β P n,β p 1 cos θ = J Nθ k= c k θ N k p 1 J 1 Nθ k= d k θ N k + δ p N, θ, 4.6 where J µ is the Bessel function of order µ, c k θ and d k θ are some coefficients and δ p N, θ is the error term. The asymptotic expansion 4.6 holds uniformly in θ, π and the coefficients can be calculated explicitly. The first coefficients are given by c θ = θ 1 sin θ 1 and d θ =. Thus, for p = 1 the asymptotic expansion equals P n,β cos θ = cos θ β θ sin θ J Nθ + δ 1 N, θ, 4.7 where the error term can be estimated by δ 1 N, θ Λ N J Nθ + J 1 Nθ, Λ >. 4.8 Here, the constant Λ is independent of θ, n and β. Now we are prepared to establish the estimation from below of the asymptotics of the coefficients ψ n,k n,k N to determine moment conditions. Notice that φ 1, φ, φ 3 1, 1 is necessary for Φ 1 z 1, z = 1 φ 1 z 1 φ z φ 3 z 1 z 1 H, see Corollary 3.11 and Basu and Reinsel [1], Proposition 1. Lemma 4.3. Let φ 1, φ, φ 3 1, 1 and at least two coefficients not equal to zero. Then there is a constant C > and x > 1 such that { fx := n, k N : ψ φ 1,φ,φ 3 n,k 1 x} C log x, x > x > 1. Proof. We consider the three different cases i φ 1 = or φ =, ii φ 1 φ and φ 3 φ 1 1 φ 1 1 and iii φ 1 φ and φ 3 φ 1 1 φ 1 < 1. Firstly, observe for φ 1 φ by 4.5 n ψ φ 1,φ,φ 3 n k n,k = φ n 1φ k 1 + φ 3 j. j j φ 1 φ j= 19

20 Thus, in the case ii φ 1 φ and φ 3 φ 1 1 φ 1 [ 1, we have ψ φ 1,φ,φ 3 n,k φ 1 n φ k for all n, k N and therefore for some x > 1 and C > fx { n, k N : φ 1 n φ k x } C log x, x x, where the inequality follows easily. Secondly, assume in the case i φ 1 = or φ = without restricting the generality φ 1 = the case φ = follows then by symmetry. By equation 4.4, the coefficients ψ n,k n,k N are given by k φ k n φ n ψ n,k = n 3, k n,, k < n. Therefore we estimate fx = {n, k N, k n : 1 k φ k n φ 3 n x} n {n, k N, k n : min φ, φ 3 φ k+n x} 1 {n, k N : min φ, φ 3 φ k+n x} C log x, for all x > x > 1 and some C >. At last, consider the case iii φ 1 φ and φ 3 φ 1 1 φ 1 < 1 and define z := φ 1 φ 1 φ , 1 and θ, π by the equation cos θ = z. Define further A β x := {k N : ψ φ 1,φ,φ 3 k+β,k 1 x}. Then the function f satisfies fx β= Aβ x. Notice that the equality ψ φ 1,φ,φ 3 n,k = φ n 1φ k ψ 1,1, n,k holds. Hence by Theorem 4. we can express the set A β x as { A β x = n N : φ 1 n+β φ n ψ 1,1, φ 3 } φ 1 φ n+β,n 1 x { = n N : φ 1 n+β φ n φ 3 n φ 1 φ P n,β φ } 1φ 1 1 x. φ 3 In the following we shall estimate φ 3 n φ 1 φ P n,β φ 3 φ 1 φ φ 1φ φ 3 1 from below using the asymptotic expansion 4.7 with error term 4.8. Hence we have P n,β cos θ = cos θ θ β sin θ J Nθ + δ 1 N, θ, β N. 4.9

21 Denote 3δ = minθ, π θ. Then if n + β+1 θ π 4 k Z π + kπ δ, π + kπ + δ, then n+1+ β+1 θ π 4 k Z π +kπ δ, π β+1 +kπ+δ. Hence the cosine term cosn+ θ π 4 can be estimated from below by cos π + δ for at least every second n N for fixed β N. Now by the error estimate 4.8 and the asymptotic formula for the Bessel function see [18], equation J µ z = πz cosz µπ π Oz, z, we can conclude that there are M N and C 1 >, such that for every fixed β N we have for at least every second n N such that N = n + β + 1/ M, the estimate θ sin θ J θ Nθ + δ 1 N, θ sinθπnθ cosnθ π 4 Λ N J Nθ + J 1 Nθ C 1 Nθ 3 1 ɛ θ cos π sinθπn θ + δ, ɛ, Therefore equations 4.9 and 4.1 yield for some ɛ > φ 3 n φ 1 φ P n,β φ 1φ 1 ɛ, φ 3 for at least every second n M and for every β N. Using these estimations it follows that for some ɛ > { A β x n N, n + β + 1 > M : φ 1 n+β φ n xɛ} Altogether we conclude the estimation in 4.11 is only done for at most all β N fulfilling β < logxɛ/ log φ 1, x > ɛ 1, thus finitely many times fx 1 { k 1, k N, k 1 k : φ 1 k 1 φ k xɛ } CM C logxɛ, x > ɛ 1, for some C, C > and the statement of the lemma follows easily. Theorem 4.4. Let φ 1, φ, φ 3,,. Then the spatial autoregressive model 4.1 admits a causal solution Y t1,t t1,t Z if and only if i the polynomial Φz 1, z 1 = 1 φ 1 z 1 φ z φ 3 z 1 z has no zero on D, and ii if at least two coefficients of φ 1, φ and φ 3 are not equal to zero, then E log + Z <, otherwise E log + Z <. If those conditions hold and P Z is symmetric, then the unique symmetric causal solution is given by Y t = k N ψ k Z t k, t Z, where k 1 ψ k = j= k j k1 + k j k φ k 1 j 1 φ k j φ j 3, k = k 1, k N

22 Proof. Suppose first that at least two coefficients of φ 1, φ, φ 3 are not equal to zero. Then sufficiency of conditions i and ii as well as the representation 4.1 follow from Theorem 3.9, since Φ 1 z 1, z = ψ k z k, z D. k N d Conversely, suppose that Y t t Z is a causal solution of 4.1. The necessity of condition i follows by Corollary It remains to show the necessity of condition ii. As in the proof of Theorem 3.7 we define the symmetrizations Ỹt t Z and Z t t Z. Then Ỹt t Z admits the representation Ỹt = k N ψ k Zt k, where k 1 ψ k = j= k j k1 + k j k φ k 1 j 1 φ k j φ j 3, k = k 1, k N, and the convergence is almost surely in the rectangular sense. By Theorem 3.3 and Corollary 3.5 we can conclude ψk P Zt k > ɛ <, ɛ > k N But equation 4.13 implies finite second log-moment, because of the following estimates E log + Z k=1 log k+1p k Z < k + 1 k=1 logk + 1 P k Z < k + 1. The last series is finite, if and only if k=1 log kp k Z < k + 1 converges. By Lemma 4.3 the latter series can be estimated from above by k=1 log kp k Z < k + 1 C =C {k Z : ψ k 1 k} P k Z < k + 1 k=1 {k Z : ψ k 1 k 1, k]} P Z k k=1 k N P ψ k Zt k 1 <, where the last inequality follows by equation Notice that the finite second logmoment of the symmetrization Z implies finiteness of the second log-moment of Z. It remains to consider the case, when two coefficients are equal to zero. In this case the model reduces to a one dimensional model. That is also the case, if φ 1 = φ = by defining the operator B = B 1 B. The model lives only on the diagonal t = t, t Z, t Z. Thus, the results [5] for the time series model can be applied, which yields necessity and sufficiency of E log + Z < and condition i in that case. An equivalent condition for i describing the parameter regions is given in Proposition 1 of Basu and Reinsel [1].

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