ALFREDAS RA CKAUSKAS AND CHARLES SUQUET The present contribution studies (I) in the general case of a separable Banach space B. The history of this pr

Transcription

1 Georgian Mathematical Journal Volume 8 (001), Number?,?{? H OLDER VERSIONS OF BANACH SPACE VALUED RANDOM FIELDS ALFREDAS RA CKAUSKAS AND CHARLES SUQUET Abstract. For rather general moduli of smoothness, like (h)=h ln (c=h), we consider the Holder spaces H (B) of functions [0; 1] d! B where B is a separable Banach space. We establish an isomorphism between H (B) and some sequence Banach space. With this analytical tool, we follow a very natural way to study, in terms of second dierences, the existence of a version in H (B) for a given random eld. 000 Mathematics Subject Classication: 60F05, 60G05, 60G17, 60G15. Key words and phrases: Banach valued Brownian motion, central limit theorem, Schauder decomposition, second dierence, skew pyramidal basis. 1. Introduction Let (), 0 1 be a modulus of smoothness and B a separable Banach space. Denote by H (B) = H ([0; 1] d ; B) the space of all functions x : [0; 1] d! B such that kx(t + h) x(t)k B = O (jhj) (1.1) uniformly in t [0; 1) d. H o (B) = H o ([0; 1] d ; B) is the subspace of functions for which O (jhj) can be replaced by o (jhj) in (1.1). Equipped with the related Holder norm (precise denitions are given in Section ), H (B) becomes a non-separable Banach space with H o (B) as a closed separable subspace. Letting the modulus vary gives a very natural scale of spaces allowing us to classify by their global regularity the functions more than continuous. This functional framework is interesting in the theory of stochastic processes since very often the continuous stochastic process under study has a better regularity than the bare continuity. Moreover, for obvious topological reasons, weak convergence in Holder spaces is stronger than in the classical space C([0; 1] d ; B) of B valued continuous functions on [0; 1] d. In the previous papers [11], [10], the authors discussed the following problem in the special case B = R: (I) For a given real valued random eld indexed by [0; 1] d ; nd sucient conditions for the existence of a version with sample paths in H o (B). ISSN X / $8.00 / c Heldermann Verlag

2 ALFREDAS RA CKAUSKAS AND CHARLES SUQUET The present contribution studies (I) in the general case of a separable Banach space B. The history of this problem began with the well known Kolmogorov's condition: if the real valued stochastic process (t), t [0; 1] satises P j(t + h) (t)j > c h 1+" ; (1.) where c; " > 0 and > 1 are constants, then it has a version with almost surely continuous paths. This was the root of successive generalizations leading to processes indexed by an abstract parameter set T. The modern theory expresses condition for the existence of a continuous version in terms of the geometry of T, i.e., some metric entropy or majorizing measure condition which estimates the size of T with respect to a pseudo metric related to. The study of Holder regularity is another branch stemming from Kolmogorov's condition. Indeed, (1.) is sucient for to have a version with sample paths in the Holder space H([0; o 1]; R) with (h) = jhj for any 0 < < "=. Ciesielski [3] gave sucient conditions for a Gaussian process to have a version with -Holderian paths. Using the method of triangular functions, Delporte [4] established sucient conditions for the existence of a version of in H([0; o 1]; R) for general moduli of smoothness. Ibragimov [6] and Nobelis [9] studied the problem (I) for general and B = R. The method of triangular functions used by Ciesielski [, 3], Delporte [4], Kerkyacharian and Roynette [7] relies on the following well known decomposition of a real valued continuous function x: x(t) = X j;v (x) j;v (t); t T; (1.3) where j;v 's are the Faber{Schauder triangular functions and V j is the set of dyadic numbers of level j in T = [0; 1]. In fact the triangular functions form a basis (in Schauder's sense) of H o ([0; 1]; R) when (h) = jhj. Moreover the Holder regularity of a continuous function is characterized by the rate of decreasing of its coecients j;v (x) in this basis. This provides the Ciesielski [] isomorphisms between these Holder spaces and some sequence spaces. These isomorphisms give a very convenient discretization procedure in [10] to study (I). In [11] we use the same method, replacing the basis of triangular functions by the basis of skew pyramidal functions dened on T = [0; 1] d, denoted again by j;v and indexed by the dyadic points v of level j in [0; 1] d. The scalar coecients j;v (x) are some dyadic second dierences of x. In the present contribution, we keep formally the same decomposition (1.3) of B-valued Holder functions into series of pyramidal functions. But now the pyramidal functions are scalar while their coecients are vectors lying in B. The important fact is the preservation in this new setting of the equivalence between the initial Holder norm kxk with the sequential norm kxk seq := sup j0 1 ( j ) max k j;v (x)k B : (1.4)

3 H OLDER VERSIONS OF BANACH SPACE VALUED RANDOM FIELDS 3 Of course, this cannot be the question of any Schauder basis for H o (B) in the general case since B itself does not necessarily possess such a basis. From (1.3) and (1.4) it should be clear that the control of the coecients j;v (x) is the key tool in our study. It is then relevant to state the basic asumptions in problem (I) in terms of the second dierences h(t) := (t + h) + (t h) (t) (1.5) of the random elds considered. This brings more exibility than the classical use of rst increments (for more argumentation see [10]). The paper is organized as follows. Section exposes the analytical preliminaries: expansion in a series of pyramidal functions, equivalence of norms, Schauder decomposability. In Section 3, we discuss problem (I) of the existence of Holderian versions. The systematic use of the sequential norm (1.4) enables us to provide sucient conditions whose general form may be sketched as follows. Assuming some uniform control of second dierences like we consider the series P h (t) B > r(jhj) (r); r > 0; (1.6) R(u) := u jd ( j ) : (1.7) Then the convergence of R(u 0 ) for some u 0 > 0 gives the existence of a version of in H (B), while the convergence for every u > 0 gives a version in H(B). o When (1.6) is obtained through weak p-moments, the corresponding result improves the Ibragimov's classical result for the case B = R by a logarithmic factor in the modulus [6]. When (1.6) is veried through exponential Orlicz norms, we have practical conditions precise enough to discriminate between H (B) and H(B). o For instance our conditions detect the optimal Holder regularity of B-valued Brownian motions. Examples of Ornstein{Uhlenbeck processes in c 0 and of p-stable processes in `r illustrate our general results.. Analytical Background Throughout T = [0; 1] d and R d is endowed with the norm jtj := max 1id jt i j, t = (t 1 ; : : : ; t d ) R d : Denote by B a Banach space with the norm jj jj B and by H (B) = H (T ; B) the set of B-valued continuous functions x : T! B such that! (x; 1) < 1; where kx(t) x(s)k! (x; ) := sup B t;st;0<jt sj< (js tj) and is a modulus of smoothness satisfying conditions (.1) to (.5) below. The technical conditions required for are the following, where c 1 ; c and c 3 are positive constants: (0) = 0; () > 0; 0 < 1; (.1)

4 4 ALFREDAS RA CKAUSKAS AND CHARLES SUQUET is non-decreasing on [0; 1]; (.) () c 1 (); 0 1=; (.3) Z 0 Z 1 (u) u du c (); 0 < 1; (.4) (u) u du c 3 (); 0 < 1: (.5) To show a concrete class of functions satisfying the conditions (.1) { (.5), let us dene inductively the sequence of functions `k on (u k ; 1) by `1(u) = ln u; u 1 = 1 and, for k, `k(u) = ln(`k 1 (u)); and u k such that `k 1 (u k ) = 1: Elementary computations show that for any (0; 1) and any nite sequence 1 ; : : : ; m R there exists a set of positive constants b 1 ; : : : ; b m such that the functions (h) = h my k=1 `k k (b k =h); 0 < h 1; satisfy conditions (.1) to (.5). The set H (B) is a Banach space when endowed with the norm kxk := kx(0)k B +! (x; 1): Obviously, an equivalent norm is obtained by replacing kx(0)k B in the above formula by kxk 1 := supfkx(t)k B ; t T g: Dene H o (B) = H o (T ; B) := fx H (B) : lim!0! (x; ) = 0g: Then H o (B) is a closed subspace of H (B): Now let us remark that for any function satisfying (.1) and (.5) there is a positive constant c 4 such that () c 4 ; 0 1: (.6) Hence the spaces H o (B) always contain all the Lipschitz B-valued functions and in particular the (continuous) piecewise ane functions. When B is itself separable, the separability of the spaces H o (B) follows by interpolation arguments. Since we are interested in the analysis of these spaces in terms of second dierences of the functions x; our rst task is to establish the equivalence of the norm kxk with some sequential norm involving the dyadic second dierences of x. Our main reference for this part is Semadeni [14]. The so-called skew pyramidal basis was introduced by Bonic, Frampton and Tromba [1] and, independently, by Ciesielski and Geba (see the historical notes in [14, p. 7]). The reader is referred also to our previous contribution [11] for a more detailed explanation. If A is a convex subset of T, the function f : T! B is said to be ane on A if it preserves the barycenter, i.e., for any nite sequence u 1 ; : : : ; u m in A and nonnegative scalars r 1 ; : : : ; r m such that P m i=1 r i = 1, f( P m i=1 r i u i ) = P m i=1 r i f(u i ).

5 H OLDER VERSIONS OF BANACH SPACE VALUED RANDOM FIELDS 5 To explain the construction of the skew pyramidal basis, dene rst the standard triangulation of the unit cube T = [0; 1] d. Write d for the set of permutations of the indexes 1; : : : ; d. For any = (i 1 ; : : : ; i d ) d ; let (T ) be the convex hull of d+1 points 0; e i1 ; (e i1 +e i ); : : : ; dp k=1 e ik ; where e i 's are the vectors of the canonical basis of R d. So, each simplex (T ) corresponds to one path from 0 to (1; : : : ; 1) via the vertices of T and such that along each segment of the path, only one coordinate increases while the others remain constant. Thus T is divided into d! simplexes with disjoint interiors. The standard triangulation of T is the family T 0 of simplexes f (T ); d g. Next, we divide T into jd dyadic cubes with edge j. By dyadic translations and change of scale, each of them is equipped with a triangulation similar to T 0. And T j is the set of jd d! simplexes so constructed. For j 1 the set W j := vert(t j ) of vertices of the simplexes in T j is W j = fk j ; 0 k j g d : In what follows we put V 0 := W 0 and V j := W j n W j 1 : So V j is the set of new vertices born with the triangulation T j. More explicitly, V j is the set of dyadic points v = (k 1 j ; : : : ; k d j ) in W j with at least one k i odd. The T j -pyramidal function j;v with peak vertex v V j is the real valued function dened on T by three conditions: i) j;v (v) = 1; ii) j;v (w) = 0 if w vert(t j ) and w 6= v; iii) j;v is ane on each simplex in T j. Observe that the notation j;v is somewhat redundant and could be simplied in v since V j 's form a partition of the set of dyadic points of [0; 1] d. It follows clearly from the above denition that the support of j;v is the union of all simplexes in T j containing the peak vertex v: By [14, Prop ], the functions j;v are obtained by dyadic translations and changes of scale: j;v (t) = ( j (t v)); t T; v V j ; from the same function with support included in [ 1; 1] d : (t) := max 0; 1 max t i <0 The B-valued coecients j;v (x) are given by 0;v (x) = x(v); v V 0 ; jt i j max t i ; t = (t1 ; : : : ; t d ) R d : t i >0 j;v (x) = x(v) 1 x(v ) + x(v + ) ; v V j ; j 1; where v and v + are dened as follows. Each v V j admits a unique representation v = (v 1 ; : : : ; v d ) with v i = k i = j ; (1 i d). The points v = (v 1 ; : : : ; v d ) and v + = (v + 1 ; : : : ; v d + ) are dened by ( ( v vi i = j if k i is odd; v if k i is even; i + vi + = j ; if k i is odd; if k i is even. v i v i

6 6 ALFREDAS RA CKAUSKAS AND CHARLES SUQUET Since v is in V j ; at least one of k i 's is odd, so v ; v and v + are really three distinct points of T. Moreover, we can write X v = v j e(v); v + = v + j e(v) with e(v) := e i ; so j;v (x) is a second dierence directed by the vector e(v). Observe further that if v is in V j, then v and v + are in W j 1. Note here that the sequences ( j;v ) and ( j;v ) are biorthogonal in the following sense. Lemma 1. For j; j 0 0 and v V j, v 0 V j 0, j;v ( j 0 ;v0) = v;v 0; where v;v 0 = 0 if v 6= v 0, v;v 0 = 1 if v = v 0 (and then j = j 0 ). Proof. Suppose rst j positive, so j;v ( j 0 ;v0) = j 0 ;v 0(v) 1 j 0 ;v0(v ) + j 0 ;v0(v+ ) ; with v V j and v + ; v W j 1. Case 1 : j < j 0. Then v, v + and v are the vertices of the triangulation T j and hence also of T j 0, but none of them can be equal to v 0, so j 0 vanishes at ;v0 v; v + ; v. Case : j > j 0. Then the segment [v ; v + ] ( with middle-point v) is contained in some simplex of T j 0 and j is ane on, so 0 j;v( j ;v0 0 ;v0) = 0. Case 3 : j = j 0. Then v and v + are in W j 0, so 1 j vanishes at v 0 and ;v0 v + and j;v ( j ;v0) = 0 j 0 ;v 0(v) = v;v0, by i) and ii) in the denition of pyramidal functions. To complete the proof, note that in the special case j = 0, 0;v ( j ;v0) = 0 j ;v0(v) = 0 v;v 0, since v is a vertex of T 0 and hence of T j 0. As usual, the space C(T ; B) of continuous functions x : T! B is endowed with the uniform norm jjxjj 1 = sup tt jjx(t)jj B : Dene the operators E j (j 0) on the space C(T ; B) by E j x := jx X i=0 vv i i;v (x) i;v ; x C(T ; B): k i odd Lemma. The B-valued function E j x is ane on each simplex of T j and such that E j x(w) = x(w) for each w W j : Proof. Since each simplex of T j is included in one simplex of T i for i j, the rst claim follows clearly from the fact that i;v 's are ane on the simplexes of T i. We check the second claim by induction on j. First, for w W 0, we have by the denition of 0;v 's and i), ii) of the denition of the pyramidal functions, E 0 x(w) = X vv 0 x(v) 0;v (w) = x(w):

7 H OLDER VERSIONS OF BANACH SPACE VALUED RANDOM FIELDS 7 Next assuming the interpolation property true for j, consider, for w in W j+1, the decomposition E j+1 x(w) = E j (w) + X +1 j+1;v (x) j+1;v (w): If w already belongs to W j, then E j x(w) = x(w) by the induction hypothesis and the second term in the above decomposition vanishes by ii) since w belonging to vert(t j+1 ) n V j+1 cannot be a peak vertex for one of j+1;v. To complete the proof, it remains to treat the case w V j+1. Using again i) and ii) we have X +1 j+1;v (x) j+1;v (w) = j+1;w (x) = x(w) 1 x(w ) + x(w + ) : Recall that w = 1 (w+ +w ). By [14, Lemma 3.4.8] there is at least one simplex in T j having w + and w as vertices. Since E j x is ane on and w and w + are in W j, we have E j x(w) = 1 and so E j+1 x(w) = x(w). Ej x(w ) + E j x(w + ) = 1 x(w ) + x(w + ) ; Since the pyramidal functions are not B-valued, they cannot form a Schauder basis of C(T ; B) as in the real valued case. Nevertheless we keep the same type of decomposition. Proposition 1. Each x in C(T ; B) admits the series expansion x(t) = X j;v (x) j;v (t); t T; where the convergence holds in the strong topology of B and is uniform with respect to t on T. Proof. For xed t there is a simplex in T j containing t. Let v 0 ; v 1 ; : : : ; v d be the vertices of. Writing! 1 () for the modulus of continuity and recalling that the diameter of is j, we have kx(t) E j x(t)k B! 1 ( j ) + kx(v 0 ) E j x(t)k B : Using the barycentric representation t = P d i=0 r i v i and Lemma, we get kx(v 0 ) E j x(t)k B = The conclusion follows. dx B r i x(v0 ) x(v i ) i=0 dx i=0 r i! 1 ( j ) =! 1 ( j ): Now we are in the position to state the equivalence of norms we were looking for. For any function x : [0; 1] d! B the (possibly innite) sequential seminorm is dened by kxk seq := sup j0 1 ( j ) max k j;v (x)k B :

8 8 ALFREDAS RA CKAUSKAS AND CHARLES SUQUET Observe moreover that when x is continuous, kxk seq = 0 if and only if x = 0. In this case, kxk seq is a (possibly innite) true norm. Proposition. Under conditions (:1) to (:5), the norm kxk is equivalent on C(T ; B) to the sequential norm, i.e., there are positive constants a; b such that for every x C(T ; B), a kxk seq kxk b kxk seq ; with nite values of the norms if and only if x H (B). Proof. The proof is exactly the same as for [11, Prop. 1], replacing jx(t)j by kx(t)k B. Note that in this proof the continuity of x was used through its expansion in a series of pyramidal functions which is now replaced by Proposition 1. Conditions (.4) and (.5) are not required for the inequality a kxk seq kxk which holds true with a = min 1; (1) for the most general moduli of smoothness. Remark. It is worth noticing that kx E J xk seq := sup j>j 1 ( j ) max k j;v (x)k vv B j is non-increasing in J. The next result is the key tool for the problem of existence of Holderian versions for a given B-valued random eld. Proposition 3. Let = ( j;v ; j 0; v V j ) be a B-valued tree and consider the following conditions. (a) (b) sup j0 (c) lim J!1 sup j>j max k j;v k B < 1. 1 ( j ) max k j;v k vv B < 1. j 1 ( j ) max k j;v k vv B = 0. j Dene the sequence (S J ) J0 of continuous piecewise ane functions by S J := JX X j;v j;v : Then under (a), S J converges in C(T ; B) to some function S. Under (b), the same convergence holds and S belongs to H (T ; B). Under (c), S J converges in H o (T ; B). Proof. The pyramidal functions being non-negative, we have j;v j;v (t) X max k j;v k B j;v (t) B X

9 H OLDER VERSIONS OF BANACH SPACE VALUED RANDOM FIELDS 9 for every t T. Recalling that 0 P j;v (t) 1 (cf., e.g., Lemma in [11]), we see that (a) entails the uniform convergence on T of S J. Under (b), the same estimate gives the convergence in C(T ; B) of S J to a function S, provided that P j0 ( j ) < 1. But this follows from our general assumption (.4), comparing series with integral. Observe now that by Lemma 1, j;v (S J ) = j;v if j J, so by the continuity on C(T ; B) of j;v 's, j;v (S) = j;v for every j; v. Applying now Proposition to the continuous function S, we obtain ksk < 1. Suppose nally that satises (c) which is obviously stronger than (b). Then S J converges (at least) in C(T ; B) sense to S H (B) and j;v (S) = j;v. Thus condition (c) means that ks S J k seq goes to zero, which gives the convergence of S J to S in the H (B) sense by Proposition. Moreover, S J 's are in H(B) o which is a closed subspace of H (B), and so S H(B). o The usefulness of Proposition 3 for the existence of Holderian versions comes from its following obvious corollary. Corollary 1. Let x be any function T! B and dene = ( j;v (x); j 0; v V j ). Then x coincides at the dyadic points of [0; 1] d with a function S which is in C(T ; B) under (a), in H (B) under (b) and in H o (B) under (c). 3. Banach Valued Random Fields with Holderian Versions We now consider a given B-valued random eld = ((t); t T ), continuous in probability, and discuss the problem of existence of a version of with almost all paths in H(B). o In our setting, a natural candidate for such a version is of the P form ~ := P j j;v () j;v. Indeed, combining Corollary 1 with continuity in probability reduces the problem to the control of maxima of random coecients k j;v ()k B which are norms of dyadic second dierences of : We dene here the second dierence h(t) in a symmetrical form by where h(t) := (t + h) + (t h) (t); t T; h C t ; C t := fh = (h 1 ; : : : ; h d ); 0 h i min(t i ; 1 t i ); 1 i dg: Theorem 1. Let = f(t); t T g be a B-valued random eld, continuous in probability. Assume there exist a function : [0; 1]! R +, (0) = 0 and a function : (0; 1]! R +, (1) = 0 such that for all r > 0, t T; h C t, Put for 0 < u < 1, P h (t) B > r(jhj) (r): (3.1) R(u) = R( ; ; ; u) := u jd ( j ) : If R(u 0 ) is nite for some 0 < u 0 < 1, then has a version in H (B). If R(u) is nite for every 0 < u < 1, then has a version in H o (B).

10 10 ALFREDAS RA CKAUSKAS AND CHARLES SUQUET When is non-increasing and non-decreasing, the same conclusions hold if R(u) is replaced by I(u) := Z 1 0 u (s) ds s d+1 : Proof. By Corollary 1 and continuity in probability, has a version in H (B) if (and only if) kk seq is nite almost surely, which is equivalent to Using the obvious bound P (kk seq lim M!1 P (kkseq > M) = 0: > M) X P k j;v ()k B > M( j ) ; (3.) this convergence follows easily from the dominated convergence theorem provided that the right hand side of (3.) be nite for some M = M 0 > 0. Recalling that Card V j (1 d ) jd and using (3.1), this last requirement is satised as soon as R(M 0 ) is nite. Invoking again Corollary 1 and continuity in probability, we see that has a version in H(B) o if and only if k E J k seq goes to zero almost surely. But this sequence being decreasing, this is equivalent to its convergence in probability. Clearly, this convergence holds if for every " > 0, X P k j;v ()k B > "( j ) < 1; which in turn follows from R(") < 1. When is non-increasing and non-decreasing, a comparison between series and integral using (3) allows us to replace R(u) by I(u) in the above estimates. Remarks on the optimality. From the very elementary nature of the proof it is easy to understand why the results are optimal in some sense. Indeed the only gaps in the proof between sucient and necessary conditions are in the use of (3.) and (3.1). But (3.) is nothing else than the majorization of the probability of a denumerable union by the sum of probabilities. So if we do not know anything on the dependence structure of these events (which is the case since assumptions on involve only its three dimensional distributions) this bound cannot be improved in general. Concerning (3.1), it is clear that for reasonable we can always construct such that (3:1) becomes an equality when reduced to t [ j V j, t h = t. Example. Let f(t); t Rg = f k (t); t Rg 1 k=1 be a sequence of independent Gaussian stationary processes. For each k 1, let r k denote the covariance P function of f k (t); t Rg. Assume that 1 expf "=r k (0)g < 1 for each " > 0. Then for each t R, the sequence (t) = f k (t)g 1 k=1 is a.s. in c 0 (see Vakhania k=1

11 H OLDER VERSIONS OF BANACH SPACE VALUED RANDOM FIELDS 11 [15]). We consider the problem when for a xed interval [a; b] the process f(t); t [a; b]g has a version satisfying certain Holder condition. Set Assume that for each " > 0 k=1 j=1 k(h) = r k (h) 4r k (h) + 3r k (0) : j k ( j ) expf " ( j )= k( j )g < 1: (3.3) Then by Theorem 1 the process f(t); 0 t 1g admits a version in H([0; o 1]; c 0 ). To prove this claim, we shall check the conditions of Theorem 1. For all r > 0; t [0; 1] and h; 0 h 1 t we have P jj h(t)jj c0 > r(h) P j h k (t)j > r(h) : (3.4) Since h k (t) has a normal distribution with mean zero and variance k(h), we have k=1 P (j h k (t)j > ) p k (h) expf = kg; > 0: (3.5) Now the conditions of Theorem 1 are satised by (3.3), (3.4) and (3.5), so the result follows. As a special case, assume that for each k 1 the process f k (t); t Rg is the Ornstein{Uhlenbeck process dened as a stationary solution of the stochastic dierential equation d k (t) = k k (t)dt + ( k ) 1= dw k (t); where fw k (t); t Rg 1 k=1 are independent Wiener processes. In this case we have r k (h) = ( k = k ) expf k jhjg. Hence, if for each " > 0 X k expf "= k g < 1; then the process f(t); t [a; b]g has a version in H o ([a; b]; c 0 ), with (h) = (hj log hj) 1= ). It is easy to consider some weaker conditions like for each " > 0 X k expf " 1 = k g < 1; where 0 < 1, which yields a weaker Holder regularity of the Ornshtein- Uhlenbeck process f(t); t Rg. The next theorem is a straightforward application of Theorem 1. When B = R, it allows a comparison with classical Kolmogorov's and Ibragimov's results.

12 1 ALFREDAS RA CKAUSKAS AND CHARLES SUQUET Theorem. Let = f(t); t T g be a B-valued random eld, continuous in probability. Assume there exist a function : [0; 1]! R +, (0) = 0 and a constant p > 0 such that for all r > 0, t T; h C t, Suppose that P h (t) B > r p (jhj) r p : (3.6) R(1) := jd p ( j ) p ( j ) < 1: Then has a version ~ in H o (B). Moreover, there is a constant C such that P (k ~ k > u) Cu p for each u > 0. Proof. Applying Theorem 1 with (r) = r p, the functional R(u) is written here jd p ( j ) R(u) := p ( j ) u p whose niteness does not depend on the value of u > 0. This explains why (3.6) cannot provide a version ~ in H (B) for some such that ~ does not belong to the corresponding H(B). o The tail behavior of k k ~ results clearly from the same behavior of kk seq which, in turn, follows from P (kk seq > u) R(u) = p R(1)u p : Remark. If (3.6) is replaced by the stronger assumption E h (t) p < B p (jhj); (3.7) p then a straightforward estimate of E kk seq gives E k k ~ p < 1. Corollary. Suppose satises (3:6) with p (jhj) = C jhj d+ for some positive constants C > 0, 0 < < p. Then has a version ~ in the Holder space H o ([0; 1] d ; B), where (jhj) = jhj =p ln (a= jhj) and > 1=p. A comparison with Corollary in the paper [6] by Ibragimov is natural here. He obtained (in the case B = R) the existence of a version in H (R) under (3.7) with p > 1 and the integral condition Z 1 0 (u) du < 1: (u)u1+d=p For the scale of moduli (jhj) = jhj ln (a= jhj) and p (jhj) = C jhj d+ provides a version of in H(R), o where (h) = h =p ln (a=h)) and > 1. this

13 H OLDER VERSIONS OF BANACH SPACE VALUED RANDOM FIELDS 13 Example. Let f(t); t Rg = f k (t); t Rg 1 k=1 be a sequence of independent symmetric p-stable stationary processes. Let, for each k 1, the process f k (t); t Rg has the stochastic representation f k (t); t Rg = Z E f k (t; u)m k (du); t R where (E; E) is a measurable space, M k is a symmetric p-stable random measure with -nite spectral measure m k. If Z k=1 E jf k (t; u)j p m k (du) < 1; then ( k (t); k 1) is a.s. in `r for each r > p: Set p k(t; h) = Z E jf k (t + h; x) + f k (t h; x) f k (t; x)j p m k (dx) and assume that there exists a function such that sup 0t1 k=1 p k(t; h) (h) and j p ( j ) p ( j ) < 1: Then the process f(t); t [0; 1]g has a version in H([0; o 1]; `r) for r > p. Indeed, since f k k (t)g 1 k=1 is a stable random element in `r with exponent p and spectral measure m = P 1 k=1 ek k ; where (e k ) is the coordinate basis in `r; we have (see, e.g., Samorodnitsky and Taqqu [13]) Z P (jj > ) c p jjxjj p`rm(dx) X 1 = C p p h(t)jj`r k(h): `r Hence the result follows by Theorem 1. Recall that a Young function is a convex increasing function on R + such that (0) = 0 and lim t!1 (t) = 1: If Z is a real valued random variable such that E(jZj =c) < 1 for some c > 0, then its -Orlicz norm is kzk := inffc > 0 : E (jzj =c) 1g: When Z is B-valued, its -Orlicz norm is dened similarly, replacing jzj by kzk B. Using the denition of the Orlicz norm, the continuity of and Beppo-Levi's theorem, it is easy to see that E (kzk B = kzk ) 1, from which we get P (kzk B > r) f(r= kzk )g 1. Theorem 3. Let be a Young function. Assume that the B-valued random eld = ( t ; t T ) is continuous in probability and satises the condition: for each t T and h C t ; If for some u > 0, k=1 jj h(t)jj (jhj): (3.8) jd u j < 1; (3.9) ;

14 14 ALFREDAS RA CKAUSKAS AND CHARLES SUQUET then admits a version with almost all paths in H (B). If (3:9) holds for each u > 0, then has a version in H o (B). Proof. Under (3.8), Theorem 1 applies with (r) = 1=(r). Of course, the scope of Theorem 3 covers the case of strong p-moments (p > ) with (3.8) replaced by (3.7). But its main practical interest is in the case of exponential Orlicz norms as in Corollaries 3 and 4 below. Then the convergence of series (3.9) can be checked through the following routine test. Lemma 3. Let be of the form (r) = c exp( br ) for some positive constants b, c,. Consider the series R(u) = u jd ( j ) and put L(j) := ( j ) j 1= ( j ) : (i) R(u) converges for some 0 < u 0 < 1 if and only if lim inf j!1 L(j) > 0; (ii) R(u) converges for every 0 < u < 1 if and only if lim j!1 L(j) = 1. The proof is elementary and shall be omitted. Observe that the qualitative form of the result does not depend on d. This parameter is involved only in the explicit determination of u 0. Corollary 3. If the B-valued random eld is continuous in probability and satises ( 3.8) for the Young function (r) = exp(r ) 1 ( > 0) and (jhj) = jhj (0 < < 1), then has a version in H o (T ; B), where (h) = jhj ln 1= (a= jhj). This improves our previous result [11, Th. 11]. The case of Gaussian random elds is of special interest. Corollary 4. Assume that the Gaussian B-valued random eld = ( t ; t T ) is continuous in probability and satises the condition: for each t T and h C t ; (i) If lim inf j!1 E h (t) B (jhj): (3.10) ( j ) j 1= ( j ) > 0, then admits a version in H (B). ( j ) (ii) If lim j!1 j 1= ( j ) = 1, then has a version in Ho (B). If for some u > 0, jd exp u ( j ) < 1; (3.11)

15 H OLDER VERSIONS OF BANACH SPACE VALUED RANDOM FIELDS 15 then admits a version in H (B). If (3:11) holds for each u > 0, then has a version in H o (B). Proof. By (3.10) and the classical estimate of the tail of Gaussian random element in the Banach space B (see, e.g., inequality (3.5) p. 59 in Ledoux Talagrand (1991)), P ( h (t) r B > r) 4 exp ; 8 (jhj) from which the result follows. Example. Let B be a separable Banach space and Y a centered Gaussian random element in B with distribution. A B-valued Brownian motion with parameter is a Gaussian process indexed by [0; 1], with independent increments such that (t) (s) has the same distribution as jt sj 1= Y. Hence (3.10) holds with (h) q = h 1= E 1= ky k B (h 0). Choosing the modulus of smoothness (h) = h ln(e=h), we see that ( j ) lim j!1 j 1= ( j ) = 1 > 0: E 1= ky k B Hence by Corollary 4 (ii), qthe B valued Brownian motion has a version in H ([0; 1]; B), where (h) = h ln(e=h); h > 0. This result cannot be improved because of Levy's theorem on the modulus of uniform continuity of the standard Brownian motion. Remark. One may ask what happens with Problem (I) when is a general modulus of smoothness, i.e., simply non-decreasing on [0; 1] and continuous at 0. In this case we can always replace the inequality kxk b kxk seq of Proposition by kxk 3 1 ( j ) max k j;v (x)k vv B =: kxk`1 : j This approach was adopted in the pioneering paper by Delporte [4] in the case d = 1, B = R. Because the norm kxk`1 is clearly weaker than kxk seq, the price paid for this greater generality is the loss of precision in the topology. For instance, in the situation of Corollary, this gives the same results as in [6]. Acknowledgement This research was supported by the cooperation agreement CNRS/LITHUANIA (4714). References 1. R. Bonic, J. Frampton, and A. Tromba, -manifolds. J. Functional Analysis 3(1969), 310{30.. Z. Ciesielski, On the isomorphisms of the spaces H and m. Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8(1960), 17{.

16 16 ALFREDAS RA CKAUSKAS AND CHARLES SUQUET 3. Z. Ciesielski, Holder conditions for realizations of Gaussian processes. Trans. Amer. Math. Soc. 99(1961), 403{ J. Delporte, Fonctions aleatoires presque s^urement continues sur un intervalle ferme. Ann. Inst. Henri-Poincare Section B 1(1964), 111{ R. V. Erickson, Lipschitz smoothness and convergence with applications to the central limit theorem for summation processes. Ann. Probab. 9(1981), 831{ I. Ibragimov, On smoothness conditions for trajectories of random functions. Theor. Probab. Appl. 8(1984), 40{6. 7. G. Kerkyacharian and B. Roynette, Une demonstration simple des theoremes de Kolmogorov, Donsker et Ito-Nisio. C. R. Acad. Sci. Paris Serie I 31(1991), 877{ M. Ledoux and M. Talagrand, Probability in Banach spaces. Springer-Verlag, Berlin, Heidelberg, Ph. Nobelis, Fonctions aleatoires lipschitziennes. Lecture Notes in Math. 850(1981), 38{ A. Rackauskas and Ch. Suquet, Central limit theorem in Holder spaces. Probab. Math. Statist. 19(1999), No. 1, 133{ A. Rackauskas and Ch. Suquet, Random elds and central limit theorem in some generalized Holder spaces. B. Grigelionis et al. (Eds.), Prob. Theory and Math. Statist., Proceedings of the 7th Vilnius Conference (1998), 599{616, TEV Vilnius and VSP Utrecht, 1999, 1. A. Rackauskas and Ch. Suquet, On the Holderian functional central limit theorem for Banach space valued i.i.d. random variables. Pub. IRMA Lille 50-III, to appear in Proceedings of the Fourth Hungarian Colloquium on Limit Theorems of Probability and Statistics, G. Samorodnitsky and M. S. Taqqu, Stable non-gaussian random processes. Stochastic models with innite variance, Chapman & Hall, Z. Semadeni, Schauder bases in Banach spaces of continuous functions. Lecture Notes in Math. 918, Springer Verlag, Berlin, Heidelberg, New York, N. N. Vakhania, Probability distributions on linear spaces. North-Holland, New York/Oxford, 1981; Russian original: Metsniereba, Tbilisi, Authors' addresses: (Received.0.001) Alfredas Rackauskas Vilnius University Naugarduko 4, LT-006, Vilnius, Lithuania Institute of Mathematics and Informatics Akademijos 4, LT-600, Vilnius, Lithuania alfredas.rackauskas@maf.vu.lt Charles Suquet Universite des Sciences et Technologies de Lille Statistique et Probabilites E.P. CNRS 1765 B^at. M, U.F.R. de Mathematiques F Villeneuve d'ascq Cedex, France Charles.Suquet@univ-lille1.fr