2.2 Moving Averages of a White Noise

Transcription

1 22 MOVING AVERAGES OF A WHITE NOISE Moving Averages of a White Noise Given the white noise fu t ; t 2 Zg, with u t 2 L 2 ; F; P/, let H u D spu t ; t 2 Z/: By definition a white noise process has zero mean Thus the condition k D covu t ; u t k / D u t u t k Eu t / 2 D, for k, is identical to u t? u t k D, for k, so that the sequence u t = u is an orthonormal sequence in H u By Proposition 14, if x 2 H u, x D a xk u k ; a xk D x u k ; (24) u 2 where P 1 ja xkj 2 < 1 (the use of a xk u k, instead of a xk u k is only a convenience, leading to the usual expression for moving averages; see below) Conversely, given the square-summable sequence a k, the series P 1 a ku k converges in H u Calling y its limit we have a k D y u k The function W u H u! `21; 1/ defined as x/ Df u a xk ; k 2 Zg; (25) is an isomorphism of H u onto l 2 1; 1/ (see Proposition 15) Definition 22 Given a square-summable sequence fa k ; k 2 Zg, the process x t D a k u t k ; t 2 Z: (26) is called a moving average of u t As in (13) the coefficients are independent of t Here however the moving average is in general infinite By (121), the autocovariance function of x t is easily seen to be s D u 2 a k a k s : Exercise 21 If the coefficients a s are real the autocovariances k are real irrespective of whether the white noise u t is real or not The converse is false Consider the process x t D u t C 1 u t 1 C u t 2 C 2u t 3 C/;

2 48 CHAPTER 2 MOVING AVERAGES where is any complex number such that j j < 1 Prove that x t is white noise, so that (1) its autocovariance function is real even when the coefficients of the moving average are not real, (2) a non trivial moving average of a white noise can be a white noise We will need the following definition Definition 23 The processes w t and z t,on ; F; P/, are costationary if both are weakly stationary and w t z t s D Ew t z t s / does not depend on t When w t and z t are costationary we set s wz D w t z t s (Show that Ez t w t s / D wz s, and is therefore independent of t, so that we can set s zw D z t w t s ) Given the moving average y t D P 1 b ku t k ; the processes x t, as defined in (26), and y t are costationary For, applying (121), it is easily seen that xy s D 2 u a k b k s : Conversely, suppose that fz t ; t 2 Zg belongs to H u Applying (24), but omitting z in the coefficients for simplicity, z t D a tk u k : (27) Now assume that z t and u t are costationary This implies that Ez t u t s / D a t;s t does not depend on t Defining b s D a t;s t, so that a tk D b kct, (27) can be rewritten as z t D a tk u k D b kct u k D b h u t h : In conclusion: hd 1 Proposition 21 Moving averages of u t, as defined in (26), belong to H u and are costationary with each other, and with u t in particular Conversely, if fz t ; t 2 Zg belongs to H u and is costationary with u t, then z t is a moving average of u t Example 25 Consider the process w t D u 2t Obviously w t is a white noise blonging to H u However, u w D u 2 whereas u t w t D u t u 2t D for t Thus w t and u t are not costationary, so that w t cannot be represented as a moving average of u t Other examples are y t D u t, and z t D u e it :

3 22 MOVING AVERAGES OF A WHITE NOISE 49 Note that given x 2 H u there exists a moving average of u t whose range (see p 3) contains x To see this consider again (24) and define z t D P 1 a xku t k, so that x D z Obviously z t is not the only moving average containing x; taking w t D z t 1 we would have y D w 1 However, it is the only moving average such that x corresponds to the integer (check this) Thus H u is covered by the ranges of fy t ; t 2 Zg, y t being any moving average of u t As an exercise, the reader may prove that if y t and z t are moving averages of u t then either the range of fy t ; t 2 Zg and the range of fz t ; t 2 Zg are equal, or their intersection is empty, and that the first alternative holds if and only if z t D y t s for an integer s Figure 21 provides a stylized picture of the spaces and subsets we are dealing with The space H u, which is mapped onto l 2 1; 1/ by the isomorphism, is partitioned into subsets, each constituting the range of a moving average of u t In the figure the range of u t itself and that of z t are represented and denoted by Œu t and Œz t respectively Different ranges do not intersect and the union of all ranges coincides with H u Let us underline again that many different processes produce the same range, like z t and w t D z t 1 Œu t Œz t x H u l 2 1; 1/ L 2 ; F; P/ FIGURE 21 A very important statement regarding processes that are moving averages of a white noise is the following Proposition 22 Let u t be a white-noise process, x t D P 1 kd k be the autocovariance function of x t Then: 1 a ku t k, and let (i) lim s!1 j s jd: (ii) j s j < j jdx 2: PROOF Without loss of generality assume that s Let a Dfa k ; k 2 Zg and put a h/ Dfa k h ; k 2 Zg Using a notation similar to that introduced in Section 138, if b 2 l 2 1; 1/, let ( b k if k h b fhg k D if k > h

4 5 CHAPTER 2 MOVING AVERAGES Note that b fhg c c fhg / D, for all b and c, and that b fhg n/ D bfh ng : Letting m be the smallest integer greater or equal to s=2, we have, using the Cauchy-Schwartz inequality and the observations above, a a s/ D Œa fmg C a a fmg / Œa fmg s/ C a s/ a fmg D a fmg a fmg s/ C a afmg / a s/ a fmg s/ / D a fmg a fm sg C a a fmg / a s/ a fmg s/ / kak ka fm sg kcka a fmg /k : s/ / Since a is square summable, given >, we can choose s such that if s > s, the tails a fm sg and a a fmg are smaller than in modulus Statement (i) follows 2kak from the observation that s D u 2a a s//: Regarding (ii), suppose that j h = jd1for an integer h > Then it easily seen that x t h D h x t ; and therefore x t sh D s h s x t D sh x t ; so that j sh jdj j, which contradicts statement (i) Very often a formal proof is much more complicated than the underlying idea For example, statement (i) in Proposition 22 can be completely grasped with the aid of Figure 22, the dotted line representing a, the starred line a 13/, so that m D 7 As one immediately realizes, when s is large enough, the belly of a corresponds to the left tail of a s/, while the belly of a s/ corresponds to the right tail of a The larger s the smaller the norm of the tails a a 13 a FIGURE 22 As we have seen, there exist weakly stationary processes belonging to H u that are not moving averages of u t Some of them however are moving averages of other white noises (as a matter of fact, some of the processes in Example 25 are white noise) In general, when a weakly stationary process z t is not defined as the moving average of a white noise, we may ask the question whether there exists a

5 22 MOVING AVERAGES OF A WHITE NOISE 51 process w t D P 1 a kv t k, with v t white noise on some probability space, such that the autocorrelation functions of z t and of w t coincide Proposition 22 provides a necessary condition for a positive answer Thus, for example, the answer is negative for the process z t D Ae it (see Example 112), whose autocovariance function does not decay for s!1 Observation 22 Dropping the condition Eu t / D from the definition of white noise has the consequence that the infinite moving average with coefficients a k makes sense only if P m kd m a k converges (as we have seen in Observation 17, the mean of P m kd m a ku t k should converge to the mean of the limit) This is a stricter condition as compared to square summability; for example, the sequence a k D ( 1 k C 1 for k for k < is square summable but not summable, see Observation 14 Observation 23 A consequence of Eu t / D is that all the variables in H u have zero mean As on p 43, denote by 1 the function of L 2 ; F; P/ which associates the real number 1 with every! 2 (the unit constant) Then define HL u D sp1; u t ; t 2 Z/ Note that Eu t / D implies that 1? u t for all t The space LH u is covered by the ranges of the moving average processes x t D c C a k u t k ; where a k is any square summable sequence and c any real number Summary Given a white noise u t, the space H u D spu t ; t 2 Z/ is isomorphic with l 2 1; 1/ H u is partitioned into subsets that are the ranges of moving averages of u t The autocovariance function s of a moving average of a white noise tends to zero as s!1