Randomly Weighted Series of Contractions in Hilbert Spaces

Transcription

1 Math. Scand. Vol. 79, o. 2, 996, ( ) Prerint Ser. o. 5, 994, Math. Inst. Aarhus. Introduction Randomly Weighted Series of Contractions in Hilbert Saces G. PESKIR, D. SCHEIDER, M. WEBER Conditions are given for the convergence of randomly weighted series of contractions in Hilbert saces. It is shown that under these conditions the series converges in oerator norm outside of a (universal) null set simultaneously for all Hilbert saces and all contractions of them. The conditions obtained are moreover shown to be as otimal as ossible. The method of roof relies uon the sectral lemma for Hilbert sace contractions (which allows us to imbed the initial roblem into a setting of Fourier analysis), the standard Gaussian randomization (which allows us further to transfer the roblem into the theory of Gaussian rocesses), and finally an inequality due to Fernique [3] (which gives an estimate of the exectation of the remum of the Gaussian (stationary) rocess over a finite interval in terms of the sectral measure associated with the rocess by means of the Bochner theorem). As a consequence of the main result we obtain: Given a sequence of indeendent and identically distributed mean zero random variables fz k g k defined on (; F; P ) satisfying EjZj 2 <, and > =2, there exists a (universal) P -null set 3 2 F such that the series: Z k () T k k converges in oerator norm for all 2 n 3, whenever H and T is a contraction in H. is a Hilbert sace The urose of the aer is to investigate and establish conditions for the convergence in oerator norm of the randomly weighted series of contractions in Hilbert saces: (.) W k () T k where fw k g k is a sequence of indeendent mean zero square-integrable random variables defined on the robability sace (; F; P ), and T is a (linear) contraction in the Hilbert sace H, while f k g k is a non-decreasing sequence of non-negative integers, and 2. Our main aim is to find sufficient conditions (and in this context to rove that they are as otimal as ossible) for the convergence of the series in (.) which is valid simultaneously for all Hilbert saces H and all contractions T in H. More recisely, we find conditions (see (3.) in Theorem 3. and (3. ) in Remark 3.2) in terms of the numbers k and EjW k j 2 for k, under which there exists a (universal) P -null set 3 2 F such that the series in (.) converges in oerator norm for all 2 n 3, whenever H is a Hilbert sace and T is a contraction in H. (This is the main result of the aer.) Then we secialize and investigate this result when W k = Z k =k AMS 98 subject classifications. Primary 4A5, 4A3, 47A6, 6F25. Secondary 42A2, 42A24, 46C, 6G5, 6G5. Key words and hrases: Randomly weighted series, indeendent, contraction, Hilbert sace, the sectral lemma for Hilbert sace contractions, Lévy s inequality, Gaussian randomization, Gaussian rocess, searable, continuous in quadratic mean, stationary, the sectral measure, the sectral reresentation theorem, orthogonal stochastic measure, the covariance function, the Bochner theorem, the Herglotz theorem, nonnegative definite, the dilation theorem of Sz.-agy, Kolmogorov s three series theorem, Szidon s theorem on lacunary trigonometrical series. goran@imf.au.dk

2 for k and >, where fz k g k is a sequence of indeendent and identically distributed mean zero random variables satisfying EjZj 2 < (see Theorem 3.4), and in articular we obtain that for > =2 the series: Z (.2) k () T k k converges in oerator norm for all 2 n 3, whenever H is a Hilbert sace and T is a contraction in H (see Corollary 3.5). In the end (see Remark 3.6) it is shown that the conditions obtained throughout are as otimal as ossible. The method of roof may be described as follows. First, we use the sectral lemma for Hilbert sace contractions (Lemma 2.), and in this way imbed the initial roblem about the series in (.) into a setting of Fourier analysis (see [4] and [7]), which concerns exressions of the form: (.3) W k e k i for some W;... ; W 2 R with... being integers for. Second, by erforming a standard rocedure of Gaussian randomization we imbed the roblem about (.3) into the theory of Gaussian rocesses. The Gaussian rocess which aears as the roduct of the randomization rocedure is given by: W k g k ( ) cos( k ) g k ( ) sin( k ) (.4) G ( ; ) = for 2 R, ( ; ) 2 g, g W ;... ; W 2 R, and... with. (Here g = fg g k k and g = fg g k k are (mutually indeendent) sequences of indeendent standard Gaussian ( (; ) ) random variables defined on ( g ; F g ; P g ) and ( g ; F g ; P g ) resectively.) The roblem in this context is reduced to estimate the exression: (.5) E. jg j For this, we aly an inequality due to Fernique [3] which estimates (.5) in terms of the sectral measure associated with the rocess G = fg g 2R by means of the Bochner theorem (see Lemma 2.2). It turns out that the sectral measure can be found exlicitly, thus allowing to obtain an accurate estimate of (.5) which suffices for our uroses. The otimality of the conditions deduced is roved by using Kolmogorov s three series theorem. To conclude the introduction, we find it convenient to mention that we are unaware of similar results, and to the best of our knowledge the sort of series which aears in (.) and (.2) has not been studied reviously. It should be also remarked that our emhasis is on the method of roof which seems to be flexible enough to admit alications in the study of various roblems having a similar character. 2. Preliminary facts In this section we find it convenient to recall and dislay some results and facts which will be used in the roof of the main result of this aer (Theorem 3.). 2

3 We begin by recalling a useful fact (called the sectral lemma for Hilbert sace contractions) which allows us to transfer the norm oerator roblems into the setting of Fourier analysis. For this, we should first clarify that a linear oerator T ( defined in a (comlex) Hilbert sace H ) is called a contraction, if kt (f )k kf k for all f 2 H (see [2]). Given a contraction T in H and f 2 H, denote P n (f ) = T n (f ); f for n, and ut n (f ) = P n (f ) for n <. Then the sequence fp n (f )g n2z is non-negative definite P P z k l k z l P kl (f ), for all z k 2 C with jkj and, see [5] (.94-95). Thus by the Herglotz theorem [5] there exists a finite ositive measure f on B(] ; ]) ( called the sectral measure of f ) such that: (2.) T n (f ); f = Z e in f (d) for all n. From this fact, by induction in degree of the olynomial, one might deduce the following remarkable inequality. (This roof was communicated to us by M: Wierdl.) Lemma 2. (The sectral lemma) If T is a contraction in a Hilbert sace H, and f is an element from H with the sectral measure f, then the inequality is satisfied: Z =2 (2.2) P (T )(f ) P (e i ) 2 f (d) whenever P (z) = P k= a kz k is a comlex olynomial of degree. Proof. The roof is carried out by induction on the degree of the olynomial P (z). If =, then we have equality in (2.2), since it follows from (2.) that kf k 2 = f (] ; ]). Suose that the inequality (2.2) is true for. Denote P P Q(z) = a k z k and R(z) = a k z k. Then by (2.) we find: (2.3) kp (T )(f )k 2 = k af Q(T )(f ) k 2 = jaj 2 kf k 2 a f; Q(T )(f ) Q(T )(f ); a f kq(t )(f )k 2 = jaj 2 kf k 2 R a Q(e i ) f (d) R a Q(e i ) f (d) kq(t )(f )k 2. Since T is a contraction, then by the assumtion we get: (2.4) kq(t )(f )k 2 = kt R(T) (f)k 2 kr(t)(f)k 2 R jr(e i )j 2 f (d) = R jq(e i )j 2 f (d). From (2.3) and (2.4) we conclude: kp(t)(f)k 2 R ja j 2 a Q(e i ) a Q(e i ) jq(e i )j 2 f (d) = R ja Q(e i )j 2 f (d) = R jp(e i )j 2 f (d). It should be noted from (2.4) in Lemma 2. that if T is an isometry in H ( kt(f)k = kf k for f 2 H ), then we have equality in (2.2). (It also follows more directly by (2.).) It allows us to 3

4 deduce the inequality (2.2) in a straightforward way by using the (oldest and best known) dilation theorem of Sz.-agy (see Theorem in [9] (.2)): If T is a contraction in a Hilbert sace H, then there exists a Hilbert sace K and a unitary oerator U in K such that H is a subsace of K and T n (f ) = Z(U n )(f ) for all n and all f 2 H, where Z is the orthogonal rojection from K into H. ( It remains to aly the equality in (2.2) to the isometry U, and then use the receding identity and the fact that kzk. ) However, if kt k <, then the error aearing in the estimate (2.2) may be noticeably large. To see this more exlicitly, take for instance P (z) = z n with n. Then the left-hand side in (2.2) equals kt n (f )k which is bounded by kt k n kfk, and thus arbitrarily small when n increases. On the other hand, the right-hand side in (2.2) equals kf k, thus being arbitrarily large as f runs through H. We will come back to this sort of analysis later on in section 4. Our next aim is to dislay an inequality due to Fernique [3] which is shown to be at the basis of our results in the next section. For this reason we shall first recall the (wide sense) stationary setting uon which this result relies (see []). Let G = fg g 2R be a (wide sense) stationary rocess defined on the robability sace (; F; P ) and taking values in the set of comlex numbers C. Thus, we have: (2.5) EjG j 2 < (2.6) E(G ) = E(G ) (2.7) Cov (G h ; G h ) = Cov (G ; G ) for all,, h 2 R. As a matter of convenience, we ose: (2.8) E(G ) = for all 2 R. Thus the covariance function of G is given by: (2.9) R() = E(G G ) for all 2 R. By the Bochner theorem there exists a finite ositive measure on B(R) such that: (2.) R() = Z eix (dx) for all 2 R. The measure is called the sectral measure of G. The sectral reresentation theorem states if R is continuous (which is equivalent to the fact that G is continuous in quadratic mean), then there exists an orthogonal stochastic measure Z on 2 B(R) such that: (2.) G = for all Z eix Z(dx) 2 R. The fundamental identity in this context is as follows: (2.2) E Z Z 2 '(x) Z(dx) = whenever ' : R C belongs to L 2 (). Hence we find: '(x) 2 (dx) 4

5 Z (2.3) E G G 2 = 2 cos()x (dx) for all, 2 R. The result may be now stated as follows. Lemma 2.2 (Fernique) Let G = fg g 2R be a stationary mean zero Gaussian rocess with values in R and the sectral measure. Suose that G is searable and continuous in quadratic mean. Then there exists a (universal) constant K > ( which does not deend on the Gaussian rocess G itself ), such that the inequality is satisfied: (2.4) E G K Z (x2^ ) (dx) =2 Z ]e x2 ;[) =2 dx Proof. It follows from (.4.5) in Theorem.4 in [3] uon taking c = b = and c k = b k = in (.4.4) for k 2. In order to make use of the inequality (2.4) we shall denote I k = [k; k] for k 2 Z. Let I R be any bounded interval, then I [ k2a I k for some finite A Z. Let K I denote the number of elements in A. Then by the searability, stationarity, and symmetry of the Gaussian rocess (2.5) E jg j 2I G = fg g 2R, we easily obtain: = E K I max k2a 2Ik jg j EjG j 2E k2a E G 2Ik. jg j = K I E. We will aly Lemma 2.2 with (2.5) in the roof of Theorem 3. (next section) to the following Gaussian rocess (obtained by the randomization rocedure described below): g k ( ) cos( k ) g k ( ) sin( k ) (2.6) G ( ; ) = W k with 2 R, ( ; ) 2 g, g W ;... ; W 2 R, and... with. Here g = fg g k k and g = fg g k k are (mutually indeendent) sequences of indeendent standard Gaussian ( (; ) ) random variables defined on the robability saces ( g ; F g ; P ) and g ( g ; F g ; P ) g resectively. It is easily verified that the orthogonal stochastic measure associated with G = fg g 2R is given by: jg j (2.7) Z ( ; ); = i W k g k ( ) W k g k ( ) fkg () fkg () =2 fkg () fkg () =2 for ( ; ) 2 g g and 2 B(R). From (2.2) and (2.7) it follows easily by indeendence that the sectral measure of G is given by: 5

6 (2.8) () = 2 jw k j () fk g fkg =2 () for 2 B(R). Having this exression in mind, one might observe that the right-hand side of inequality (2.4) (alied to the Gaussian rocess (2.6)) gets a rather exlicit form which is easily estimated in a rigorous way. This will be demonstrated in more detail in the roof of Theorem 3.. In the remaining art of this section we describe the rocedure of Gaussian randomization. It allows us to imbed the roblem under consideration (which involves exressions (.3) and aears by alications of (2.2)) into the theory of Gaussian rocesses. It will be used below in the roof of our main result (Theorem 3.). It should be observed that the Gaussian rocess in (2.6) aears recisely after erforming Gaussian randomization as described in the next lemma. Lemma 2.3 (Gaussian randomization) Let W = fw k g k be a sequence of indeendent mean zero random variables defined on the robability sace (; F; P ), let g = fg g k k and g = fg g k k be (mutually indeendent and indeendent from W ) sequences of indeendent standard Gaussian ( (; ) ) random variables defined on the robability saces ( g ; F g ; P g ) and ( g ; F g ; P g ) resectively, and let k be non-negative integers for k. Then the inequality is satisfied: (2.9) E k W k e i 8 E W k g k cos( k) g k sin( k) for all. ( It should be clarified here that the sequences W, g and g, initially defined on, g and g resectively, might be well-defined as the rojections onto the first, second and third coordinate of the roduct robability sace ( g g; F F g F g ; P P g P g ), thus being mutually indeendent. The second exectation sign in (2.9) denotes the P P g P - g integral. Moreover, this robability sace is to be enlarged in the roof below as follows. The Rademacher sequence " = f" k g k ( " k s are indeendent and take values 6 with robability =2 ) aearing in the roof below, initially defined on the robability sace ( " ; F " ; P " ), is understood to be defined as the rojection onto the fifth coordinate of the roduct robability sace ( g g " ; F F g F g F F " ; P P g P g P P " ), thus being indeendent of W, g, g and W, where W = fw k g k is an indeendent coy of W defined on ( ; F ; P ) and thus, as the rojection onto the fourth coordinate, indeendent from W, g, g and ".) Proof. First note that by the triangle inequality we have: (2.2) E k W k e i E E W k cos( k ) W k sin( k ). In the remainder, as a matter of convenience, we shall write cs instead of either cos or sin function. Let W = fw k g k be an indeendent coy of W defined on ( ; F ; P ), and let " = f" k g k be a Rademacher sequence defined on the robability sace ( " ; F " ; P " ) and 6

7 understood to be indeendent from both W and W. Then f" k (W k W )g k k and fw k W g k k are identically distributed for any given and fixed choice of signs " k = 6 with k, and since W k s are of mean zero, we get: (2.2) E EE W k cs( k ) = E (W k W k )cs( k ) = EE (W k E (W k )) cs( k) " k (W k W k )cs( k. ) Taking the P " -integral on both sides in (2.2), and using the triangle inequality, we obtain: (2.22) E W k cs( k ) 2 EE " " k W k cs( k ) On the other hand, since f" k jg k j g k and fg k g k are identically distributed, we have: (2.23) EE " = " k W k cs( k ) r r = r 2 E E "E g 2 EE g 2 EE " ow from (2.2), (2.22) and (2.23) we easily conclude: E W k e k i 2 EE g 2 EE g 2 2 EE g E g " k jg k j W k cs( k ) g k W k cs( k ) This is recisely (2.9), and the roof is comlete. g k W k sin( k ) " k E g (jg k j) W k cs( k ). g k W k cos( k ). g k W k cos( k ) g k W k sin( k ). For convenience of the reader we shall conclude this section by recalling the Lévy s inequality (for roof see [6].47-48) which is used in the roof of our main result in the next section. Lemma 2.4 (Lévy s inequalities) Let = f k g k be a sequence of indeendent and symmetric random variables with values in a searable Banach sace B. Denote P n S n = i= i for all n. Then we have: n o (2.24) 8 9 P max kn ks kk > t 2P ks n k > t 7

8 (2.25) E max ks k k kn 2E ks n k for all t >, all < <, and all n. 3. The main results In this section we resent the main results of the aer. We begin by the next fundamental theorem which should be read together with Remark 3.2 following it. In the subsequent art of this section we ass to a more elaborate analysis of this result. Theorem 3. Let W = fw k g k be a sequence of indeendent mean zero random variables defined on the robability sace (; F ; P ), and let f k g k be a non-decreasing sequence of non-negative integers ( with > ). Suose that there exist integers := n < n < n 2 <..., such that the following condition is satisfied: (3.) i= q log(n i ) ni k=n i E W k =2 2 <. Then there exists a (universal) sequence of P -integrable random variables M = fmg defined on (; F; P ) which converges to zero P -a.s. and in P -mean such that for any Hilbert sace H and any contraction T in H we have: (3.2) R R>n k=n W k () T k M () for all 2 and all. In articular, there exists a (universal) P -null set 3 2 F such that the series: (3.3) W k () T k converges in oerator norm for all 2 n 3, whenever H is a Hilbert sace and T is a contraction in H. Proof. Given R > n for some, there exists (a unique) l R such that: (3.4) n lr < R n lr. Let f 2 H be given and fixed, and let f be the sectral measure of f. Then by the triangle inequality and the sectral lemma (see (2.2) in Lemma 2.) we get: (3.5) R W k () (f) T k R W k () (f) T k k=n k=n lr lr i= i= n i W k () (f) T k k=n i n i W k () (f) T k k=n i 8

9 j= n j<rn j Z i= j= i= j= n i k=n i R k=n j Z n j <Rn j W k () T k (f ) W k () e i k 2 f (d) n i k=n i n j<rn j R k=n j =2 W k () e i k kf k W k () e i k 2 f (d) R k=n j =2 W k () e ik kf k for all 2. In order to control the last term in (3.5), we shall generate W = fw k g k an indeendent coy of W = fw k g k and aly Lévy s inequality (2.25) with = to the sequence of indeendent and symmetric random variables f (W k W k ) ei k g k with values in the searable Banach sace B of all bounded continuous comlex valued functions on ] ; ] with resect to the -norm. In this way we obtain: (3.6) E W k e i k k=n j R = E W k E (W k ) e i k n j <Rn j k=n j R EE Wk W n j <Rn j k e i k k=n j 2EE 4E n j <Rn j n j k=n j n j k=n j R Wk Wk e i k W k e i k for all j. Taking remums in (3.5) first over all f 2 H with kf k and then over all R > n, it is clear that the roof of (3.2) will be comleted as soon as we show that: (3.7) i= E ni W k e k i k=ni ote that once (3.7) being roved, the variables M (3.8) M () = i= <. ni W k () e k i k=ni aearing in (3.2) may be defined as follows: 9

10 j= n j <Rn j R k=n j W k () e k i for all 2 and all. The roof of (3.7) is carried out in two stes. First, by erforming the standard Gaussian randomization (see (2.9) in Lemma 2.3) we get: (3.9) E 8 E n i W k e k i n i W k k=n i k=n i g k cos( k) g sin( k k) for all i, where g = fg g k k and g = fg g k k are (mutually indeendent and indeendent from W ) sequences of indeendent standard Gaussian ( (; ) ) random variables defined on ( ; F ; P ) and ( ; F ; P ) resectively. g g g g g g Second, we aly Lemma 2.2 to the stationary Gaussian rocess: G ( ; ) = n i k=n i for 2 R and ( ; ) 2 g g by (2.5) and (2.8) we get: (3.) E 2K n i k=n i ni W k () g k ( ) cos( k ) g k ( ) sin( k ), where 2 and i are given and fixed. Thus W k () g k cos( k) g k sin( k ) 8 2 =2 jw k ()j 2 k ^ k=ni Z n i k=ni E n i k=n i W k () g k jw k ()j 2 f y j k >ex(y 2 )g (x) =2 dx where K > is the (universal) numerical constant from (2.4) not deending on the given and fixed 2 and i. The first term on the right-hand side of the inequality in (3.) is easily controlled by Jensen s inequality and indeendence: (3.) E ni k=ni W k () g k E ni k=ni W k () g k 2 =2 n i = k=ni jw k ()j 2 =2 To estimate the last term on the right-hand side of the inequality in (3.), we shall again use Jensen s inequality. In this way we get: (3.2) Z ni =2 jw k ()j 2 f y j k >ex(y 2 )g (x) dx k=ni q Z log(n i ) ni log( ni ) k=ni =4 ni = log( ni ) k=ni jw k ()j 2 f y j k>ex(y 2 )g (x) jw k ()j 2 log k =2 dx log( ni ) =2

11 for the given and fixed 2 (3.3) E n i and i. From (3.), (3.) and (3.2) we obtain the estimate: W k () g k cos( k) g k sin( k) k=n i ni q 8 (2K) log(n i ) k=n i jwk()j 2 =2 ni k=n i jwk()j 2 =2 for the given and fixed 2 and i. ow, taking the P -integral in (3.3) and alying Jensen s inequality to the second term on the right-hand side, we see by (3.) and (3.9) that (3.7) holds. From this, by (3.6) and (3.8), we see that E(M ) < for all, and hence from the secific form of M s we conclude M P -a.s. and in P -mean as. This comletes the roof of (3.2). The last statement about the series in (3.3) follows clearly from (3.2). Remark 3.2 The roof of Theorem 3. shows (see (3.2) above) that condition (3.) might be weakened to the following condition: (3. ) i= log( ni ) =4 E ni k=n i log Wk =2 k 2 <. Under (3. ) we have again (3.2) and (3.3). In this context it should be observed that (3.) follows straightforward from (3. ) by Jensen s inequality. While aarently weaker as stated in (3. ), the condition seems to be most convenient in the form given in (3.). Remark 3.3 P In the context of condition (3. ) in Remark 3.2 it is worthy to recall that i= jx ij 2 =2 P i= jx ij whenever x i 2 R for i. Hence we see that if the conditions (3.) and (3. ) are aimed to go beyond the straightforward estimate (which is easily obtained by the triangle inequality): Wk kt k k kfk Wk kfk (3.4) W k T k (f ) then i 7 n i must be of a considerable growth. (The estimate (3.4) also indicates that the main emhasis of the result in Theorem 3. is on the cases where kt k = ; see also Remark 3.6 below) Moreover, if (3. ) is satisfied, then we have: q log( ni ) E W ni <. (3.5) i= This indicates that in order to aly (3.) or (3. ) we must have EjW ni j as i. Motivated by the last fact in Remark 3.3, in the remaining art of this section we investigate: (3.6) W k = Z k =r k for k, where Z k s are indeendent and identically distributed mean zero random variables satisfying EjZ j 2 <, and r k s are ositive real numbers tending to infinity as k. More

12 exlicitly, we take r k = k with >, but other choices are ossible as well. The result of Theorem 3. may be then refined as follows. Theorem 3.4 Let Z = fz k g k be a sequence of indeendent and identically distributed mean zero random variables defined on the robability sace (; F ; P ) such that EjZj 2 <, and let f k g k be a non-decreasing sequence of non-negative integers ( with > ) satisfying the following condition: q log ( ni ) (3.7) < (n i= i ) =2 for some integers n < n 2 <..., and some > =2. Then there exists a (universal) sequence of P -integrable random variables M = fm g defined on (; F; P ) which converges to zero P -a.s. and in P -mean such that for any Hilbert sace H and any contraction T in H we have: R (3.8) M () R>n k=n Z k () k T k for all 2 and all. In articular, there exists a (universal) P -null set 3 2 F such that the series: (3.9) Z k () k T k converges in oerator norm for all 2 n 3, whenever H is a Hilbert sace and T is a contraction in H. Proof. We aly Theorem 3. with W k = Z k =k for k. By the identical distribution of Z k s we see that in order to verify condition (3.) we have to estimate the exression: (3.2) i= q log( ni ) = ni E =2 Wk 2 k=n i EjZ j 2 =2 i= For this, a simle integral comarison shows that: (3.2) n i k 2 k=n i Z ni dx = 2 x 2 n i q ni log( ni ) k=n i k 2 =2. (n i ) 2 (n i ) 2 2 (n i ) 2 for all (3.22) i. Inserting (3.2) into (3.2), and using (3.7), we get: i= q log( ni ) ni k=n i E Wk 2 =2 2

13 EjZ j 2 2 =2 i= log ( ni ) (n i ) =2 <. Thus (3.) is satisfied, and (3.8) and (3.9) follow from (3.2) and (3.3). This comletes the roof. Corollary 3.5 Let Z = fz k g k be a sequence of indeendent and identically distributed mean zero random variables defined on the robability sace (; F; P ) such that EjZj 2 <, and let > =2 be a given number. Then there exists a (universal) P -null set 3 2 F such that the series: (3.23) Z k () k T k converges in oerator norm for all 2 n 3, whenever H is a Hilbert sace and T is a contraction in H. Proof. We aly Theorem 3.4 with k = k and n i = 2 i. For this, note that we have: log ( ) ni i (n i= i ) =2 = log 2 2 i= i(=2) <. Thus condition (3.7) is fulfilled, and (3.23) follows from (3.9). This comletes the roof. Remark 3.6 The conclusion of Theorem 3.4 (and Corollary 3.5) fails for < =2, unless Z k = P -a.s. for all k. Indeed, if the conclusion would be true with = =2 for instance, then uon taking T = I we would have: Z k () k T k (f ) = Z k () k kfk < for P -a.s. 2. Thus by Kolmogorov s three series theorem we would obtain: for all Var Zk k fjz kjc kg = C >. Hence (with C = ) we would easily get: Var Z fjz j kg k Var Z k fjz kjc kg < as k, so that we could conclude Z = P -a.s. The case < < =2 is treated in exactly the same manner. This comletes the roof of the claim. It should also be observed by triangle inequality that the result of Theorem 3.4 (and Corollary 3.5) follows easily either for > or kt k < ( rovided that > and k s do not tend to infinity too slowly when k ). This shows that Theorem 3.4 (with Corollary 3.5) treats essentially the cases (u to the simultaneity) where kt k = and =2 <. 3

14 4. Some remarks on the roof In the remaining art of the aer we shortly analyze otimality of the inequalities used in the roof of our main result in Theorem 3.. Our motivation for this direction relies uon two facts. Firstly, we feel that the method of roof resented is instructive and can be modified to treat similar questions. Secondly, we see from Remark 3.6 that the conditions obtained throughout are as otimal as ossible, but u to the fact that the assertions imlied are valid simultaneously for all Hilbert saces and all contractions of them. Thus the information to be given below might clarify the scoe and lead to an imrovement when something similar (in such a generality) is not required. The first inequality in (3.5) is a consequence of the triangle inequality and reflects our basic idea of assing to the blocks of random elements under consideration. Since the length of the blocks is not fixed and may vary, the error aearing in (3.5) can be made arbitrarily small. The third inequality in (3.5) may contain a noticeable large error ( when kt k< ) as already recorded following Lemma 2.. It is also mentioned there that this inequality becomes an equality if T is an isometry. Thus, in the context of our results where the convergence is to be obtained simultaneously for all contractions of Hilbert saces, the third inequality in (3.5) is as otimal as ossible. It should be ket in mind, however, that this is not the case in general. In the context of the fourth inequality in (3.5) it could be worthwhile to record the following. By Jensen s inequality (alied twice) we find: (4.) n i n i n i k=n i This can be rewritten in the form: (4.2) n i k=n i EjW k j n i n i EjW k j n i n i n i n i n i k=n i n i n i k=n i k=n i EjW k j 2 EjWk j 2=2 EjW k j 2 =2. =2. Consider moreover the case when the sequence f k g k is lacunary, which means k = k > for k with some > ( taken to be stricly less than > ). Then k > k, and thus log(k) > k log() for all k. Hence from (4.2) we easily get: (4.3) n i k=n i EjWkj q log(n i ) n i k=n i EjWkj 2 =2. The inequality (4.3) shows that condition (3.) in Theorem 3. imlies that: (4.4) E jwkj = EjWkj <. This clearly indicates that the result of Theorem 3. (and Theorem 3.4 with Corollary 3.5) does not go beyond a triangle inequality argument in the case when the sequence fkg k is lacunary. We think that this fact is by itself of theoretical interest. To exlore the Gaussian randomization inequality (3.9) (see Lemma 2.3), we could note that by 4

15 (3.27)(3.3) in [8] (together with Lévy s inequality and a assage to W k W, where k fw k g k is an indeendent coy of fw k g k, indeendent of fg k g k as well) we get the inequality: (4.5) E ni k=n i g k W k cs( k ) 32 r 2 n i n i E ni W k cs( k ) k=n i where cs stands for either cos or sin. This shows otimality of (3.9). Finally, the inequality aearing in (3.) (see Lemma 2.2) is known to be shar (see [3]). REFERECES [] ASH, R: B: and GARDER, M: F: (975). Toics in Stochastic Processes. Academic Press. [2] DUFORD, : and SCHWARTZ, J: T: (958). Linear Oerators, Part I: General Theory. Interscience Publ: Inc., ew York. [3] FERIQUE, : (99). Régularité de fonctions aléatoires gaussiennes stationnaires. Probab. Theory Related Fields 88 (52-536). [4] KAHAE, J: P: (968): Some Random Series of Functions: First edition, D: C: Heath and Comany; Second edition, Cambridge University Press. [5] KREGEL, U: (985). Ergodic Theorems. Walter de Gruyter & Co., Berlin. [6] LEDOU, M: and TALAGRAD, M. (99): Probability in Banach Saces (Isoerimetry and Processes): Sringer-Verlag Berlin Heidelberg. [7] MARCUS, M: B: and PISIER, G: (98). Random Fourier Series with Alications to Harmonic Analysis: Annals of Mathematics Studies, o:, Princeton, ew Jersey: Princeton University Press and University of Tokyo Press. [8] PESKIR, G. (994). From uniform laws of large numbers to uniform ergodic theorems. Math. Inst. Aarhus, Prerint Ser. o. 3 (8 ). Lecture otes Series o. 66, 2, Det. Math. Univ. Aarhus, (42 ). [9] SZ.-AGY, B: (974). Unitary Dilations of Hilbert Sace Oerators and Related Toics. The American Mathematical Society, Regional Conferences Series in Mathematics, o: 9. Goran Peskir Deartment of Mathematical Sciences University of Aarhus, Denmark y Munkegade, DK-8 Aarhus home.imf.au.dk/goran goran@imf.au.dk Dominique Schneider Institut de Recherche Mathématique Avancée Université Louis Pasteur et CRS 7, rue René Descartes, 6784 Strasbourg France Michel Weber Institut de Recherche Mathématique Avancée Université Louis Pasteur et CRS 7, rue René Descartes, 6784 Strasbourg France 5