Convergence results for conditional expectations

Transcription

1 Beroulli 11(4), 2005, Covergece results for coditioal expectatios IRENE CRIMALDI 1 ad LUCA PRATELLI 2 1 Departmet of Mathematics, Uiversity of Bologa, Piazza di Porta Sa Doato 5, Bologa, Italy. crimaldi@dm.uibo.it 2 Accademia Navale, Viale Italia 72, Livoro, Italy. pratel@mail.dm.uipi.it Let E, F be two Polish spaces ad [X, Y ], [X, Y] radom variables with values i E 3 F (ot ecessarily defied o the same probability space). We show some coditios which are sufficiet i order to assure that, for each bouded cotiuous fuctio f o E 3 F, the coditioal expectatio of f (X, Y )givey coverges i distributio to the coditioal expectatio of f (X, Y) give Y. Keywords: coditioal expectatio; Skorohod s theorem; weak covergece of probability measures 1. Itroductio Let E, F be two Polish spaces. Let X, Y be two radom variables defied o a probability space (Ù, A, P) with values i E, F, respectively. Moreover, for each iteger > 0, o a probability space (Ù, A, P ), let X be a radom variable with values i E ad Y a radom variable with values i F. The problem we cosider here is to fid coditios uder which, for each bouded cotiuous fuctio f o E 3 F, we have the weak covergece of the distributio uder P of the coditioal expectatio E P [ f (X, Y )jy ] to the distributio uder P of the coditioal expectatio E P ½f (X, Y )jy Š. Problems of this kid arise i the theory of filterig, which plays a fudametal role i various fields, such as mathematical fiace, biology ad telecommuicatios. Ideed, i the theory of filterig, it is kow that the coditioal expectatio of the sigal give the observatio is the optimal estimate, i the sese of the miimum mea square error. Computatio of this coditioal expectatio is, i geeral, extremely difficult ad so it is atural to seek approximatios. Thus the problem is to fid coditios uder which the approximatio of the sigal observatio pair leads to a coditioal expectatio that is close (i some sese) to the coditioal expectatio of the sigal give the observatio. A first result i this directio may be foud i Goggi (1994; 1997), where a chage of probability measure is assumed, from P to a suitable Q ad from P to a suitable Q, i such a way that, i particular, P is absolutely cotiuous with respect to Q o the ó -field ó (X, Y ), P is absolutely cotiuous with respect to Q o the ó -field ó (X, Y ), ad, for each, the radom variables X, Y are idepedet uder Q ad the radom variables X, Y are idepedet uder Q. I this paper, we replace the assumptio of idepedece by the less restrictive assumptio that, for each bouded cotiuous fuctio g o E, the distributio uder Q of E Q ½g(X )jy ] coverges weakly to the distributio uder Q of # 2005 ISI/BS

2 738 I. Crimaldi ad L. Pratelli E Q ½ g(x )jy ]. This coditio becomes ecessary if we assume that P is equivalet to Q o ó (X, Y ) (see Corollary 4.2). Moreover, we obtai a result for the covergece of the coditioal expectatios ot oly of the form E P ½f (X )jy Š (as i Goggi 1994; 1997), but also of the form E P ½f (X, Y )jy Š. This allows us (see Corollary 4.1) to obtai that, if the distributio uder P of [X, Y ] coverges weakly to the distributio uder P of [X, Y ], the the weak covergece of the distributio uder P of E P ½f (X )jy Š to the distributio uder P of E P ½f (X )jy Š for each bouded cotiuous fuctio f o E is equivalet to the weak covergece of the distributio uder P of E P ½f (X, Y )jy Š to the distributio uder P of E P ½f (X, Y )jyš, for each bouded cotiuous fuctio f o E 3 F. Fially, we would like to poit out the simplicity of our proof compared to the oe preseted by Goggi. The paper is structured as follows. We preset our mai result (Theorem 2.1) i Sectio 2 ad prove it i Sectio 3. I Sectio 4 we fid some characterizatios for the covergece of coditioal expectatios ad prove that, if the two probability measures P, Q are equivalet o ó (X, Y ), coditio (b) i Theorem 2.1 is ecessary. I Sectio 5, we show that the result give by Goggi is a particular case of our Theorem Mai result Let E, F be two Polish spaces, edowed with their Borel ó -fields. O a probability space (Ù, A, P), let X be a radom variable with values i E ad Y a radom variable with values i F. For each iteger > 0, o a probability space (Ù, A, P ), let X be a radom variable with values i E ad Y a radom variable with values i F. Let Q be a probability measure o (Ù, A) such that P is absolutely cotiuous with respect to Q o the ó -field ó (X, Y ) geerated by [X, Y ], ad let us deote by Z a versio of the correspodig Rado Nikodym derivative. Moreover, for each, let Q be a probability measure o (Ù, A ) such that P is absolutely cotiuous with respect to Q o the ó -field ó (X, Y ) geerated by [X, Y ], ad let us deote by Z a versio of the correspodig Rado Nikodym derivative. Thus, we have Z ¼ l(x, Y ), Z ¼ l (X, Y ), where l, l are suitable positive real Borel fuctios o E 3 F. Deote by Z:Q the probability measure o A which has desity Z with respect to Q. Similarly, for each > 0, deote by Z :Q the probability measure o A which has desity Z with respect to Q. We shall prove the followig result: Theorem 2.1. I the above settig, let us assume the followig coditios: (a) The distributio í of [X, Y, Z ] uder Q coverges weakly to the distributio í of [X, Y, Z] uder Q. (b) For each bouded cotiuous fuctio g o E, the distributio uder Q of the coditioal expectatio E Q ½g(X )jy ] coverges weakly to the distributio uder Q of the coditioal expectatio E Q ½ g(x)jy ].

3 Covergece results for coditioal expectatios 739 The, for each bouded cotiuous fuctio f o E 3 F, the distributio uder P of the coditioal expectatio E P ½f (X, Y )jy Š coverges weakly to the distributio uder P of the coditioal expectatio E P ½f (X, Y )jy Š. 3. Proof of Theorem 2.1 Let us start by observig that, sice we have the relatios P ¼ Z :Q o ó (X, Y ), P ¼ Z:Q o ó (X, Y ), i order to prove Theorem 2.1 we may replace the probability measures P, P by the probability measures Z:Q, Z :Q, respectively. Thus, we may work oly with the probability spaces (Ù, A, Q), (Ù, A, Q ) ad the triplets of radom variables (X, Y, Z), (X, Y, Z ). For each bouded cotiuous fuctio g o E, the distributio uder Q of the coditioal expectatio E Q ½g(X )jy ] depeds oly o the distributio of [X, Y ] uder Q. Moreover, if f is a bouded cotiuous fuctio o E 3 F, ad U, V are versios of the coditioal expectatios E Q [ZjY ], E Q [ f (X, Y )ZjY ], the a versio W of the coditioal expectatio E Z:Q ½f (X, Y )jy Š is give by W ¼ r(u, V), where r is the real Borel fuctio defied o R 2 by r(u, v) ¼ v=u, for u 6¼ 0 0, for u ¼ 0: Therefore, the distributio of W uder Z:Q depeds oly o the distributio í of [X, Y, Z] uder Q. Thus, we see that, i order to prove Theorem 2.1, we may replace the triplet (X, Y, Z) by aother oe, say (X 9, Y 9, Z9) (possibly, defied o a ew probability space), provided that its joit distributio is í. It is worthwhile to observe that, sice we require that the joit distributio of the ew triplet is the same as the old oe, we have the equality Z9 ¼ l(x 9, Y 9) almost everywhere. Similarly, for each, we may replace the triplet (X, Y, Z ) by aother oe, say (X 9, Y 9, Z9 ) (possibly, defied o a ew probability space), provided that its joit distributio is í. O the other had, assumptio (a) ad Skorohod s theorem allow us to choose the ew triplets (X 9, Y 9, Z9) ad (X 9, Y 9, Z9 )i such a way that they are defied o a commo probability space (Ù9, A9, Q9) ad, o this space, the radom variable [X 9, Y 9, Z9 ] coverges almost surely to [X 9, Y 9, Z9]. Summig up, what we have just observed allows us, without loss of geerality, to cosider oly the particular case i which all the probability spaces (Ù, A, Q ) coicide with (Ù, A, Q) ad, o this space, the radom variable [X, Y, Z ] coverges almost surely to the radom variable [X, Y, Z]. Assumig this is the case, let us observe that, by Scheffé s theorem, the sequece (Z ) coverges i L 1 (Q) toz. Now, we divide the proof ito two steps. (1)

4 740 I. Crimaldi ad L. Pratelli Step 1. Let us prove that, if g is a bouded cotiuous fuctio o E, ad T, T are versios of the coditioal expectatios E Q [g(x )jy ], E Q [g(x )jy ], the T coverges i probability to T. To this ed, let us observe that the radom variables T are uiformly bouded ad, by assumptio, the sequece (T ) coverges i distributio uder Q to T. Thus, sice we have the equality jt Tj 2 dq ¼ T 2 dq 2 TT dq þ T 2 dq, it suffices to prove that we have Ð T 2 dq ¼ lim Ð TT dq, or, more geerally, RT dq ¼ lim RT dq (2) for each bouded radom variable R which is measurable with respect to the ó -field ó (Y ) geerated by Y, that is, of the form h(y ), where h is a bouded Borel fuctio o F. O the other had, sice the fuctios of this type for which the desired covergece holds form a mootoe class, we ca limit ourselves to takig ito accout oly the case of a bouded cotiuous fuctio h o F. I this case, the assertio immediately follows: ideed, by the covergece i distributio of [X, Y ]to[x, Y ] ad the covergece i probability of Y to Y,wehave h(y )T dq ¼ h(y )g(x )dq ¼ lim h(y )g(x )dq ¼ lim h(y )T dq ¼ lim h(y )T dq: Step 2. Let f be a bouded cotiuous fuctio o E 3 F ad V, V versios of the coditioal expectatios E Q [ f (X, Y )ZjY ], E Q [ f (X, Y )Z jy ]: Let us prove that V coverges i L 1 (Q) tov. To this ed, recall that we have Z ¼ l(x, Y ) ad observe that, if we deote by ì the distributio of [X, Y ] uder Q, for each E. 0, we ca fid a iteger k ad k pairs of fuctios (g 1, h 1 ),...,(g k, h k ), where g k is a bouded cotiuous fuctio o E ad h k is a bouded cotiuous fuctio o F, such that Xk f (x, y)l(x, y) g j (x)h j (y), E, that is, Xk f (X, Y )Z g j (X )h j (Y ) L 1 (ì) L 1 (Q), E: (3)

5 Covergece results for coditioal expectatios 741 O the other had, uder Q, the sequece ( f (X, Y )Z P k g j(x )h j (Y )) coverges almost surely to the radom variable f (X, Y )Z Xk g j (X )h j (Y ): Moreover, it is uiformly itegrable: ideed, the fuctios f, g j, h j are bouded ad, as we have already observed, the sequece (Z ) coverges i L 1 (Q) toz. Therefore, the above covergece is also i L 1 (Q). Thus, by iequality (3), for sufficietly large, we have f (X, Y )Z Xk g j (X )h j (Y ) Let T j, T j, be versios of the coditioal expectatios L 1 (Q) E Q [g j (X )jy ], E Q [g j (X )jy ]: The, by Jese s iequality ad relatios (3) ad (4), we fid kv V k L 1 (Q), 2E þ Xk kt j h j (Y ) T j, h j (Y )k L 1 (Q):, E: (4) Hece, lettig go to þ1 ad usig what we have proved i step 1 ad the fact that Y coverges i probability uder Q to Y, we obtai lim supkv V k L 1 (Q) < 2E: Sice E is arbitrary, the covergece of V is proved. I particular (for f ¼ 1), it follows that, if U, U are versios of the coditioal expectatios E Q [ZjY ], E Q [Z jy ], the U coverges i L 1 (Q) to U. Thus, we have that the radom variable [U, V ] coverges to [U, V] i probability uder Q (ad so uder Z:Q). Moreover, sice we have Z dq ¼ U dq ¼ 0, fu¼0g fu¼0g the set of the discotiuity poits of the fuctio r (defied by (1)) is egligible with respect to the distributio of [U, V] uder the probability measure Z:Q. Therefore, we ca affirm that the radom variable W ¼ r(u, V ) coverges to W ¼ r(u, V) i probability (ad so i distributio) uder Z:Q. Fially, rememberig that Z coverges i L 1 (Q) toz, we fid that the distributio of W uder Z :Q coverges weakly to the distributio of W uder Z:Q. This proves the theorem sice the radom variables W, W are versios of the coditioal expectatios E Z:Q ½f (X, Y )jy Š, E Z :Q ½f (X, Y )jy Š:

6 742 I. Crimaldi ad L. Pratelli 4. Some complemets From Theorem 2.1 we obtai the followig corollaries: Corollary 4.1. Let E, F be two Polish spaces, edowed with their Borel ó -fields. O a probability space (Ù, A, P), let X be a radom variable with values i E ad Y a radom variable with values i F. Moreover, for each iteger > 0, o a probability space (Ù, A, P ), let X be a radom variable with values i E ad Y a radom variable with values i F. The the followig statemets are equivalet: (i) For each bouded cotiuous fuctio f o E 3 F, the distributio uder P of the coditioal expectatio E P ½f (X, Y )jy Š coverges weakly to the distributio uder P of the coditioal expectatio E P ½f (X, Y )jy Š. (ii) The distributio uder P of [X, Y ] coverges weakly to the distributio uder P of [X, Y ] ad, for each bouded cotiuous fuctio g o E, the distributio uder P of E P ½g(X )jy Š coverges weakly to the distributio uder P of E P ½g(X )jy Š. Proof. Implicatio (i) ) (ii) is obvious. Implicatio (ii) ) (i) is a particular case of Theorem 2.1; that is, the case i which we have P ¼ Q (ad so Z ¼ 1) ad P ¼ Q (ad so Z ¼ 1) for each. h Corollary 4.2. Uder the otatio of Theorem 2.1, let us assume coditio (a) ad QfZ. 0g ¼1 (that is, P equivalet to Q o ó (X, Y )). Moreover, let us assume that, for each bouded cotiuous fuctio f o E, the distributio uder P of the coditioal expectatio E P ½f (X )jy Š coverges weakly to the distributio uder P of the coditioal expectatio E P ½f (X )jy Š. The coditio (b) of Theorem 2.1 holds. Proof. Sice, by assumptio, the distributio of Z uder Q coverges weakly to the distributio of Z uder Q ad we have QfZ. 0g ¼1, we obtai that lim Q fz. 0g ¼QfZ. 0g ¼1: (5) Therefore, for sufficietly large, the followig radom variables are well defied: W ¼ Q fz. 0g 1 I f Z.0g: Further, we fid lim E Q (W 1) 2 ¼ 0: (6) If we set Z9 ¼ W =Z ad Z9 ¼ 1=Z, the probability measure Q9 ¼ W :Q is absolutely cotiuous with respect to P o ó (X, Y ) with Rado Nikodym derivative Z9 ad Q9 ¼ Q is absolutely cotiuous with respect to P o ó (X, Y ) with Rado Nikodym derivative Z9. It is easy to see (usig Skorohod s theorem ad Scheffé s theorem) that, by coditio (a) of Theorem 2.1 ad equality (5), we have that the distributio uder P of the radom variable [X, Y, Z9 ] coverges weakly to the distributio uder P of the radom variable [X, Y, Z9].

7 Covergece results for coditioal expectatios 743 Hece, applyig Theorem 2.1, we obtai that, for each bouded cotiuous fuctio f o E 3 F, the distributio uder Q9 of E Q9 ½f (X, Y )jy Š coverges weakly to the distributio uder Q9 of E Q9 ½f (X, Y )jy Š. I particular, we obtai that, for each bouded cotiuous fuctio g o E, the distributio uder Q9 of E Q9 ½g(X )jy Š coverges weakly to the distributio uder Q9 of E Q9 ½g(X )jy Š. Recallig that Q9 ¼ W :Q ad Q9 ¼ Q, sice equality (6) holds, we may coclude that coditio (b) of Theorem 2.1 is satisfied. h With a argumet similar to the oe used i the proof of step 1 of Theorem 2.1, we obtia the followig propositio: Propositio 4.3. I the settig of Corollary 4.1, let us assume that, for each, the probability space (Ù, A, P ) coicides with (Ù, A, P). The the followig coditios are equivalet: (i) For each bouded cotiuous fuctio f o E 3 F, the coditioal expectatio E P ½f (X, Y )jy Š coverges i L 1 (P) to the coditioal expectatio E P ½f (X, Y )jy Š. (ii) The sequece (Y ) coverges i probability to Y ad, for each bouded cotiuous fuctio f o E 3 F, the coditioal expectatio E P ½f (X, Y )jy Š coverges i distributio to the coditioal expectatio E P ½f (X, Y )jy Š. Proof. Regardig implicatio (i) ) (ii), we have oly to prove that the covergece i probability of h(y )toh(y) for each bouded cotiuous fuctio h o F is equivalet to the covergece i probability of Y to Y. To this ed, let us fix a coutable basis U of F ad, for each ope set U i U, let us deote by h U a bouded positive cotiuous fuctio o F such that fh U. 0g ¼U. Thus, if we start from a give subsequece of (Y ), it is possible to extract (by a diagoal argumet) a sub-subsequece, say (Y 9 ), such that, for each U i U, the sequece (h U (Y 9 )Þ coverges almost surely to h U (Y ). Therefore, there exists a set H i A with P(H) ¼ 1 ad such that, for each ø i H ad U i U, the sequece (h U (Y 9 (ø))þ coverges to h U (Y (ø)). The it is easy to see that, if ø belogs to H, the sequece (Y 9 (ø)) coverges to Y (ø): ideed, for each U i U with Y (ø) 2 U, sice h U (Y (ø)). 0, we have h U (Y 9 (ø)). 0, that is, Y 9 (ø) 2 U, for sufficietly large. Implicatio (ii) ) (i) follows from a argumet similar to the oe used i step 1 of the proof of Theorem 2.1. h From Corollary 4.1 ad Propositio 4.3 we obtai the followig corollary: Corollary 4.4. I the settig of Propositio 4.3, the followig coditios are equivalet: (i) For each bouded cotiuous fuctio f o E 3 F, the coditioal expectatio E P ½f (X, Y )jy Š coverges i L 1 (P) to the coditioal expectatio E P ½f (X, Y )jy Š. (ii) The sequece (Y ) coverges i probability to Y, the radom variable [X, Y ] coverges i distributio to [X, Y ] ad, for each bouded cotiuous fuctio g o E, the coditioal expectatio E P ½g(X )jy Š coverges i distributio to the coditioal expectatio E P ½g(X )jy Š.

8 744 I. Crimaldi ad L. Pratelli 5. Compariso with Goggi s result Let us use the same otatio as i the previous sectios. The result proved i Goggi (1994) is the followig: Theorem 5.1. For each, let Q be a probability measure o (Ù, A ) such that the probability measure P is absolutely cotiuous with respect to Q o the ó -field ó (X, Y ), ad let us deote by Z a versio of the correspodig Rado Nikodym derivative. Let us assume the followig coditios: (A) O the probability space (Ù, A) there exist a probability measure Q ad a positive ó (X, Y )-measurable radom variable Z with E Q [Z] ¼ 1 such that the distributio of [X, Y, Z ] uder Q coverges weakly to the distributio of [X, Y, Z] uder Q. (B) The distributio uder P of [X, Y ] coverges weakly to the distributio uder P of [X, Y ]. (C) For each, the two radom variables X,Y are idepedet uder Q ad the two radom variables X, Y are idepedet uder Q. The the followig statemets hold: (i) The probability measure P is absolutely cotiuous with respect to Q o the ó -field ó (X, Y ), ad Z is the correspodig Rado Nikodym derivative. (ii) For each bouded cotiuous fuctio f o E, we have that the distributio uder P of the coditioal expectatio E P ½f (X )jy Š coverges weakly to the distributio uder P of the coditioal expectatio E P ½f (X )jy Š. It is easy to see that we ca obtai the above theorem as a corollary of Theorem 2.1. More precisely, we have the followig corollary: Corollary 5.2. With the otatio of Theorem 5.1, let us assume coditios (A), (B) ad (C). The, the probability measure P is absolutely cotiuous with respect to Q o ó (X, Y ) with Rado Nikodym derivative Z ad coditio (b) ad so the assertio of Theorem 2.1 holds. Proof. We observe that, by coditios (A) ad (B), for each bouded cotiuous fuctio f o E 3 F, wehave f (X, Y )dp ¼ lim f (X, Y )dp ¼ lim f (X, Y )Z dq ¼ f (X, Y )Z dq (where the last equality follows by Skorohod s theorem ad Scheffé s theorem). Moreover, by coditio (C), for each bouded cotiuous fuctio g o E ad for each, wehave E Q g(x )jy ] ¼ E Q [g(x )] ad E Q g(x )jy ] ¼ E Q [g(x )]:

9 Covergece results for coditioal expectatios 745 Thus, i order to arrive at the coclusio, it is sufficiet to remember that, by assumptio, the distributio uder Q of X coverges weakly to the distributio uder Q of X. h Refereces Goggi, E.M. (1994) Covergece i distributio of coditioal expectatios. A. Probab., 22(2), Goggi, E.M. (1997) A L 1 -approximatio for coditioal expectatios. Stochastics Stochastics Rep., 60, Received February 2004 ad revised August 2004