1.4 Complex random variables and processes

Transcription

1 .4. COMPLEX RANDOM VARIABLES AND PROCESSES 33.4 Complex random variables and processes Analysis of a stochastic difference equation as simple as x t D ax t C bx t 2 C u t can be conveniently carried over by introducing complex stochastic variables and processes. Basic notions and definitions regarding the field of complex numbers C are assumed to be known. Let us just recall that given the complex number z D acib, z D a ib is called the conjugate of z, z C z is the real 2a, z z is the purely imaginary 2ib, zz D a 2 C b 2 Djzj 2, the square modulus of z (not to be confused with z 2 D.a 2 b 2 / C 2iab). Moreover, there exists one and only one angle 2. (or.02 if you prefer) such that a Djzjcos and b Djzjsin. This angle is referred to as the argument of z and denoted by z. We have z D a C ib Djzj.cos z C i sin z /: The latter is called the polar representation of z, in that the point.a; b/ of the Cartesian plane is represented by means of the polar coordinates jzj and z. A well known trigonometric formula gives and therefore zw Djzjjwj.cos. z C w / C i sin. z C w //; (.25) z n Djzj n.cos n z C i sin n z /: Thus, if z does not coincide with 0 or, the trajectory of z t for t 2 N describes a closing spiral, an opening spiral or remains on the unit circle according to whether the modulus of z is smaller, larger or equal to unity. Extension of the power function, polynomial functions, or rational functions (ratios of polynomials) to the complex field are trivial. Less trivial but not difficult is the extension of other analytic functions, that is, functions that possess a Taylor expansion. In particular, we are interested in the exponential function x! e x.it is well known that the Taylor expansion of the exponential function at 0 is e x D C! x C 2! x2 CD X jd0 j! xj ; (.26)

2 34 CHAPTER. DEFINITIONS the series converging absolutely for all x 2 R. This implies that the series C! z C 2! z2 CD X jd0 j! zj converges absolutely, and therefore converges, for all z 2 C, so that we may give the following definition which makes sense for all z 2 C. e z D C! z C 2! z2 CD X jd0 j! zj ; Exercise.9 Be sure that you remember that a series of real or complex numbers P kd0 b n converges if and only if for every >0 there exists n such that if n > n then jb n C b nc CCb ncp j < for all p (this is the Cauchy condition for series); and that if the series converges absolutely, i.e. P kd0 jb kj converges, then it converges. Now observe that the fundamental identity e xcy D e x e y ; (.27) holding for all real numbers x and y, can be obtained algebraically from (.26). For, given the series P kd0 a k and P kd0 b k, we can consider the double sequence a k b h. If both series converge absolutely to the limits A and B respectively, then any series obtained by arbitrarily grouping the terms a k b h (all of them), converges absolutely to AB (see [3, p. 204]). In our case, multiplication and polynomial grouping gives the series whose terms are! x C! y D!.x C y/ 2! x2 C! xy C 2! y2 D 2!.x C y/2

3 .4. COMPLEX RANDOM VARIABLES AND PROCESSES 35 so that (.27) is obtained in the limit. This argument can obviously be applied to the extension of the exponential to the complex field, that is, (.27) holds for x and y belonging to C. As a consequence, On the other hand, e z D e acib D e a e ib : e ib D C! ib C 2!.ib/2 C D b 2 C 2! 4! b4 C C i! b 3! b 3 C 5! b5 C Now observe that the series between brackets are the Taylor expansions of the cosine and sine function respectively, so that, in conclusion, e acib D e a.cos b C i sin b/: Note that the polar representation can be rewritten as z Djzje i z ; that the rule (.25) is easily reobtained, without resorting to any trigonometric formula, by observing that zw Djzjjwj e i z e i w Djzjjwj e i. zc w / ; : and that z n Djzj n e in z : Exercise.20 Using e ib D cos b C i sin b, derive cos b D eib C e ib ; sin b D eib e ib : (.28) 2 2i Using (.29) derive the formulas for cos.a C b/ and sin.a C b/. The complex exponential function maps C into C. We will also make use of the function t! e it, mapping, for a given real,, orr, into C (actually the range of this function is a subset of the unit circle), and of the function! e ik, mapping, for a given integer k, the interval Œ,orR, into C. Observation.5 (Periodic functions, period and frequency.) Let f W R! C.. f is periodic of period P > 0 if f.x C np / D f.x/ for all n 2. Of course if P is a period for f then np, n being a positive integer, is a period.

4 36 CHAPTER. DEFINITIONS 2. Suppose that P and Q are periods of f and Q > P. Then, f.x C n.q P// D f..x np / C nq/ D f.x np / D f.x/: Thus Q P is a period of f. 3. Suppose that there exists a minimum period P of f. Then if Q is a period of f, Q D np, with n a positive integer. For, if mp < Q <.m C /P, for a positive integer m, then Q mp would be a period. But Q mp < P. For a function without a minimum period, consider g.x/ D if x is rational, 0 if x is irrational. 4. Suppose that f has period P. The function f is periodic on if and only if one of its periods is integer, i.e. if qp D p 2 for integers p and q, i.e. if P D p=q. 5. Let f be of period P. The number D =P is the number of cycles between x and x C, per unit of time if x is interpreted as time. is called the frequency of the periodic function f. 6. The function e it has minimum period 2 (the period of the cosine and sine functions) and frequency =2. Given any real, the period of e it is 2= and the frequency =2. The number is known as the angular frequency. It represents the number (not an integer in general) of cycles (revolutions around the unit circle) made by the complex number e it while t goes from 0 to 2. The revolutions are clockwise if < 0, anticlockwise if > 0; if D 0 the complex number e it does not move from. As no confusion will arise we shall refer for brevity to as frequency instead of angular frequency, though the first term should be reserved to =2, the number of cycles per unit of time. 7. We only deal with discrete-time. Therefore the time-function e it, for a real, maps into C. As a consequence the range of can be limited to the interval. (or any other half closed interval of the same length). For, given we have D C 2k, with k integer and 2. (fairly obvious), so that e it D e i tci2t D e i t e i2kt D e i t ; as, for t integer, e i2kt D. (Of course the same conclusion would not hold for continuous time.) The definition of the vector space C n is obvious. The inner product, norm and distance are defined as v nx ux x y D x k y k ; jxj Dxx D t n jx k j 2 ; d.x; y/ Djx yj kd (note the squared moduli underneath the square-root). The inner product fulfills (P), (P3), (P4) of Section.3.7, whereas (P2) becomes kd0

5 .4. COMPLEX RANDOM VARIABLES AND PROCESSES 37 (P2 0 ) x y D y x: Note that while.ax/ y D a.x y/; x.ay/ D.ay/ x D a.y x/ D a.x y/: (.29) Exercise.2 Check that least squares, orthogonal projection, Cauchy-Schwartz inequality, triangular inequality, convergence, Cauchy criterion, can be reformulated in C n and all the results hold. Exercise.22 Define l 2. ; / on the complex field as the set of complex sequences fr k g such that P P kd jr kj 2 < : Define the inner product as v w D kd v kw k. Check that (P), (P3), (P4) and (P2 0 ) hold and all the results of Section.3.8 can be obtained in the same way. Now, given a measure space.; F; m/, where m might be a probability measure, consider all the functions mapping into C, i.e. all complex functions whose domain is. Obviously, a function f W! C determines two functions mapping into R such that f.!/ D f.!/ C if 2.!/. A function f is measurable if f and f 2 are measurable. It is integrable if f and f 2 are integrable and in that case, by definition, f.!/dm.!/ D f.!/dm.!/ C i f 2.!/dm.!/: (.30) The space L 2.; F; m/, on the complex field, is defined as the set of all functions f W! C such that jf.!/j 2 dm.!/ D.f.!/ 2 C f 2.!/ 2 /dm.!/ < (note the square modulus in the integral). Inner product and norm are defined as s f g D f.!/g.!/dm.!/; kf kd jf.!/j 2 dm.!/; (.3) the distance being defined as usual as kf gk. Henceforth L 2.; F; m/ will always denote the space of square integrable complex functions (so that heavy notation such as L 2.; F; m; C/ or L 2.; F; m; R/ will not be necessary). Exercise.23 Starting with the fact that integrals are defined by means of sums, extend the definition of integral to functions taking values in C and show that (.30) is a consequence of the definition. Observe that all we need to define the integral of

6 38 CHAPTER. DEFINITIONS a function F W! V is that V is a vector space on the real or the complex field, so that sum and multiplication by a scalar are defined, and that V has a metric, so that convergence is defined. For example, V might be a Hilbert space. Show that if V is a finite-dimensional vector space, so that F.!/ D.F.!/; F 2.!/; :::; F n.!//; the integral of F is the vector whose components are the integrals of the components of F. We will come back to this point when defining stochastic integrals in Chapter 3. Exercise.24 Check that (P), (P3), (P4) and (P2 0 ) hold for the space L 2.; F; m/, on the complex field, with the inner product defined in (.3), and that all the statements of Section.3.9 can be obtained in the same way. Quite obviously, the definition of a general Hilbert space on the complex field only requires axioms (P), (P3), (P4) and (P2 0 ). Obviously, both `2. ; / and L 2.; F; m/, on the complex field, are Hilbert spaces. All the statements in Section.3.0 hold. Note that (.2) becomes v w D X kd.v v k /.w v k /: (.32) For, consider that v w D lim m! D lim m! kd! mx m! X.v v k /v k.w v k /v k mx m kd m kd m! Œ.v v k /v k Œ.w v k /v k : The conclusion follows from (.29). Thus the map mapping v to the sequence of its Fourier coefficients establishes an isomorphism between M and the space l 2. ; / on the complex field. Lastly, note that (.20) takes the form kvk 2 D X kd jv v k j 2 : (.33) Now assume that the measure space is a probability space.; F; P/. The random variable z W! C determines two real random variables x and y such that z D x C iy. The distribution of z is defined as the distribution of the real

7 .4. COMPLEX RANDOM VARIABLES AND PROCESSES 39 vector.x; y/. Given a vector D.z ; z 2 ; :::; z n / of complex random variables on.; F; P/, with z j D x j C iy j, the distribution of is the distribution of the real vector.x ; x 2 ; :::; x n ; y ; y 2 ; :::; y n /: The variables z and z 2 are independent if P Œ.x ; y ; x 2 ; y 2 / 2 A B D P Œ.x ; y / 2 A/ P Œ.x 2 ; y 2 / 2 B/ ; where A and B are subset of R 2. Exercise.25 Show that if z D z C iz 2 and w D w C iw 2 are independent then z j and w k are independent for all j ; k D ; 2 (thus independence of z and w implies that cov.z j ;w k / D 0 for all j ; k D ; 2). The converse is not true, i.e. independence of z j and w k for all j ; k D ; 2 does not imply that z and w are independent. For, consider the following stochastic variables, where D Œ0 with the uniform distribution: a.!/ D ( for!< 2 0 otherwise 8 ˆ< if!< 4 a 2.!/ D if 2 ˆ:!<3 4 0 otherwise 8 ˆ< if!< 4 a 3.!/ D if! 3 4 ˆ: 0 otherwise Prove that a j and a k are independent where j k, j ; k D ; 3. However, a and the vector.a 2 a 3 / are not independent. Indeed whereas P.a D and.a 2 a 3 / D.// D P.a D ; a 2 D ; a 3 D / D =4; P.a D / P..a 2 a 3 / D.// D P.a D / P.a 2 D / P.a 3 D / D =8 (this example is adapted from [9], p. 95, exercise (4)). Thus if we define z D a C ia and w D a 2 C ia 3, z and w are not independent although z j and w k are independent for all j ; k D ; 2. The space L 2.; F; P/ over the complex field is the obvious specification of L 2.; F; m/ over the complex field. Note that the definition of covariance for complex random variables is h i cov.x; y/ D E.x E.x/.y E.y// D.x.!/ E.x/.y.!/ E.y/dP.!/;

8 40 CHAPTER. DEFINITIONS so that cov.x; y/ D cov.y; x/ and var.x/ D cov.x; x/ D jx.!/ E.x/j 2 dp.!/: Of course, convergence of x n to x in mean square implies convergence to zero of both the variance and the mean of x x n (see Observation.7). The definitions of complex stochastic process, stationary and weak stationary complex processes do not present any difficulty. Note that for complex weakly stationary processes we have k D cov.x t ; x t k / D cov.x t k ; x t / D cov.x t ; x tck / D k : Example.2 (Complex version of Example.3) Let A be a zero-mean complex stochastic variable belonging to L 2.; F; P/ and a real number. Define x t D Ae it : (.34) It is easy to prove that x t is a weakly stationary process: E.x t / D E.A/e it D 0; k D E Ae it Ae i.t k/ D A 2eik : Exercise.26 Let A j, j D ; 2;:::; n; be zero-mean mutually orthogonal stochastic variables belonging to L 2.; F; P/, i.e. E.A j A k / D 0. Let j, j D ; 2;:::; n; be real numbers and define x t D A e i t C A 2 e i 2t CCA n e i nt : (.35) Prove that x t is weakly stationary and determine its autocovariance function. Exercise.27 In Exercise.26 assume n D 2, A 2 D A, 2 D. Setting A D a C ib, prove that the orthogonality assumption on A? A 2, i.e. A? A, implies that var.a/ D var.b/, cov.a; b/ D 0, and that x t D 2.a cos t b sin t/; which is Example.3, up to a trivial redefinition. Exercise.28 Assume that A and are independent stochastic variables on the probability space.; F; P/, that A has zero mean, is real and define x t D Ae it : Prove that if A has finite second moment then x t is weakly stationary. Obtain Example.4 in the same way as Example.3 is obtained in Example.27. (For this example see [7], p. 479, Example 4.)

9 .4. COMPLEX RANDOM VARIABLES AND PROCESSES 4 Exercise.29 Given two complex stochastic variables z and w on.; F; P/, let z D z C iz 2 ; w D w C iw 2 ; where z j and w j, j D ; 2, are real stochastic variables on.; F; P/. Show that if cov.z j ;w k / D 0, for j ; k D ; 2, then cov.z;w/ D 0, but that the converse statement is not true. An example is w D Nz, under the conditions cov.z ; z 2 / D 0, var.z / Dvar.z 2 /. In this case zero-correlation of z and w does not imply cov.z j ;w k / D 0 for all j ; k D ; 2 (nor independence of z and w; see Exercise.25). Summary. We have briefly reviewed basic notions on complex numbers and introduced the complex function e z by extending the Taylor expansion of the real exponential function. Complex vector spaces, Hilbert spaces, the complex function space L 2, complex random variables and processes have been defined. The basic results on convergence and completeness can be proved by fairly obvious modifications of their real counterparts.

10 42 CHAPTER. DEFINITIONS