nthen for almost all (a.a.) te<0, 1),

Séria I: PRACE M A T E M A T Y C ZN E XX IX (1989)

Au g u s t

M.

(Lublin)

M a rcin k ie w icz -Z y g m u n d law o f large num bers in v ector spaces

Abstract. W e establish the la w .o f large numbers o f the Marcinkiewicz-Zygmund type for random elements in a measurable vector space. Next we apply the obtained result to derive analogous theorems for a wide class of linear metric spaces and modular spaces.

1. Introduction. Marcinkiewicz and Zygmund [3] in their joint work have proved, among other things, the following theorem: if X lt X 2, ... is a sequence of independent copies of a measurable function X x: <0, 1 }-+R , such that

l

(i) J |X1(t)|rdt < oo for some r, 0 < r < 2, and

о T

(ii) j ydF(y)->0 as Т -ю о, where F is the common distribution of X js,

- T

n

then for almost all (a.a.) te<0, 1), n~1/r £ Xj(t)-+0 as n->oo.

j = i

It turned out that the theorem was true for random variables defined on an arbitrary probability space, assumption (ii) for 0 < r < 1 could be rejected and the condition £|X1|r < oo was not only sufficient but also necessary for a.s.

П

convergence of n~1/r Xj to zero (such a version of the Marcinkie-

j = i

wicz-Zygmund law of large numbers can be found, e.g., in Loéve’s book [2], p.

243).

We shall be interested here in the case when the expectation of considered random variables does not exist. A similar situation was discussed also in a paper by Woyczynski [5], where various generalizations of the Marcin­

kiewicz-Zygmund theorem for some subclasses of separable (and complete) Fréchet spaces were obtained. We extend the Marcinkiewicz-Zygmund law to

A M S subject classifications 1980. Primary 60 В 12, 60 B 11, 60 В 05, 60 F 15, Secondary 60 G 50.

K e y w o r d s a n d p h r a s e s . M arcinkiewicz-Zygm und law o f large numbers, measurable vector space, linear metric space, modular space.

random elements taking values in a measurable vector space, and next we prove similar results for locally pseudoconvex linear metric spaces and modular spaces, in particular — for Orlicz spaces. We do not assume that measurable vector spaces or linear metric spaces are complete and the condition of separability for our purposes is inessentially too, thus our results generalize those of Woyczynski [5].

In Section 2 we quote fundamental definitions, Section 3 contains a generalization of Toeplitz’ and Kronecker’s Lemmas, and Section 4 is devoted to various criteria ensuring a.s. convergence to zero of normalized sums of random elements in a measurable vector space. In Section 5 we prove Marcinkiewicz-Zygmund type theorems for measurable vector space, and finally in Section 6 Marcinkiewicz-Zygmund type laws of large numbers for locally pseudoconvex linear metric spaces and modular spaces are given.

2. Fundamental notions and definitions. For convenience of the reader we remind here basic facts which will be used in next sections.

Let L be a real vector space with a <r-field of subsets of L and let (R, 08) denote the real line with its Borel (7-field. A pair (L, 0£) is called a measurable vector space if operations (x, y)->x + y and (a, x)->ax from (Lx L, x 0£) and (R x L, 08 x ££) respectively into (L, 0£) are measurable.

Let (Q, 3F, P) be a probability space. A random element (r.e.) X in L is a (# ”, immeasurable function X : Q->L. The measurability of addition and multiplication by scalars implies that the а-field is invariant under translations and homothecies. Therefore every real linear combination of r.e.’s in L is a r.e. in L.

Throughout the paper a function q: L->R+

<0,

(not vanishing identically) is termed as a seminorm if q(0) = 0 and q{x + y) ^ q{x) + q(y) for x, yeL .

We say that a seminorm q determines p-convex neighbourhoods of zero, 0 < p ^ 1 (or shortly q is p-convex) if for all x, yeL , arbitrary r > 0 and a, f ^ 0 such that aP + (3P = 1, q(ocx + fiy) < r whenever q(x) < r and q(y) < r. It can be noted that a p-convex seminorm q is nondecreasing in the sense that q(ax) ^ q(fix) for a fixed x e L and any real numbers ft ^ a ^ 0. Assuming in addition that q is symmetric, i.e., q(x) = q( — x) for all x e L , we have g(ax) ^ q(fix) provided 0 ^ |a| ^ \fi\.

A seminorm q is said to be linearly continuous at zero if for each x e L

(2.1) g(ax)->0 as a->0+.

A seminorm q is called p-subhomogeneous, 0 < p ^ 1 if for all a ^ 0 and x e L

(2.2) £jr(ax) ^ ccpq(x).

The foregoing relation with the inequality replaced by equality defines

a p-homogeneous seminorm q.

It can be easily seen that a p-subhomogeneous seminorm q is p-convex and linearly continuous at zero.

To avoid repetitions we make the convention that means the passage to the limit as n-+oo. Furthermore, in all the sums, definitions of sequences, triangular arrays etc., indices change as follows: к = 1, 2 ,..., kn, where kn -> oo, j = 1, 2, ..., n, and n = 1 ,2 ,... The letter в in a formula means that it should be read twice with + and — signs, e.g. £ g (0 x n>) < °o is equivalent

П to two conditions: £<?(*„) < oo and Х з ( - х „ ) <

.

n n

3. A generalization of Toeplitz’ and Kronecker’s Lemmas. In this section we generalize the well-known Toeplitz’ and Kronecker’s Lemmas. These results enable us to obtain laws of large numbers on the basis of convergence of a sequence or a series of r.e.’s in a measurable vector space L. Though we write below about the Cauchy criterion, sequences satisfying such a condition need not be convergent and completeness of the space L is not required.

Lemma 3.1.

Let {ank} be a triangular array of real numbers such that

(3.1) ДО < ank-*0

к

and

(3.2)

V V Д Z

< c where

< p ^ 1 .

m 0 < C < oo n k > m

Let q be a p-convex linearly continuous at zero seminorm in a vector space L and let x'n = Y

ankxk> where x„eL.

к

(a) If q(x„ — x)->0 and

(3.3) 0 < ^ а „ к-Ь „->0,

к

then q(x'n — bnx)-+ 0.

(b) If (x„) satisfies the Cauchy criterion with respect to q, that is

(3.4) Л

V A

e > 0 i m, n > i

and if

holds for ank

c f llpak, where an, bn>

and

then

г к

q(e{x'n- b „ x kn + i))-+ 0.

In particular, if q(xn)-+ 0, then, by (a), g(x'„)->0. Moreover, if q(xn — x)->0 and if a„k, b„ and cn are such as in (b), then q(x'n — bnx)-+ 0.

Conversely, if (a) or (b) is true for all sequences ( x j c= L satisfying our

assumptions, then

and

hold, provided left-hand side inequalities are

fulfilled.

Proof, (a) Without loss of generality we assume that C1/p is a positive integer. Given any e > 0, for к > r = max(m, ke) we have q(xk — x) < eC-1/p, so that q ( Y ank(xk — x)) ^ C1/pq(C~1/p Y апк(хк~х)) < e because q is p-convex.

k > r k > r

Moreover, since q is linearly continuous at zero, by (3.1) and (3.3) we obtain

q(Y ank(xk~ *))-»0 and q((Yank-bn)x)-^ 0, which completes the proof of (a).

k ^ r к

(b) Suppose that (3.4) is satisfied. Then on account of p-convexity of q ^and

(2.1) for sufficiently large n ^> i,

q(0c;ilp ^Y ak(xk- xi)) ^ Y # c ~ 1/pa*(xk- x f)) < £

к < i k < i

and q(Qc~1/p Y аЛхк~хд) K £- Furthermore

k ^ i

q(o(cnllpY ak-bn)xi) < £’

к and since q is nondecreasing,

д(0Ь„(*;-*лп + 1)) ^ 4 (0(х;-х*„ + 1)) < e.

To prove the converse statement, put xn = x ^and xn = x(l+<5„fc), where

q(x) > 0.

Lemma 3.2.

Let q be a p-convex linearly continuous at zero seminorm in a vector space L. If {sn = Y xj\ sa^sfies the Cauchy criterion with respect to q, and {dn, n ^ 0} is a sequence of real numbers such that 0 = d0 < dt < ... and j cn = Y(dj~dj-i)p 00» then

j

(3.5) 0-

j

In particular, if q(0(sn-x))-+0, then ^(3.5) holds.

Proof. Let s0 ^{= 0 and} an ⁼ dn ^— d„-1. ^Then

Cn 1/p Y djxi = cn llpdnsn ~cn 1/p Y aisi-1 >

j j

thus applying Lemma 2.1 (b) we find that q(6c~llPY djxj)~+ ^0.

j

Lemma 3.2 should be used only in the situation when {s„} satisfies the Cauchy criterion, but it is not known whether the series of seminorms Y^(xn)

П

does converge or not. However, in most applications of similar results to

probability theory we want to find conditions ensuring convergence of

arithmetic means expressed in terms of seminorms of considered r.e.’s. In this

case we can apply a simple lemma which essentially improves the rate of

convergence in comparison with Lemma 3.2.

L

3.3. Let {en} be a nondecreasing sequence of positive real numbers such that e„->oo and let q be a p-subhomogeneous seminorm in L. If

Z <?(*„) < 00 > then

<г(<С‘ Х > л Н о-

j

Proof. According to the Kronecker Lemma we have Ч(ей 1 Z ejXj) < e “ p Z e^q{x) -> 0.

j j

4. A.s. convergence of normalized sums of r.e.’s in a measurable vector space.

If we intend to check whether a given sequence of r.e.’s fulfils the law of large numbers the important thing is to have at the disposal a number of handy criteria which can guarantee this fact. Basing upon the results of Section 3 one can easily obtain various theorems ensuring a.s. convergence to zero of consecutive arithmetic means formed from L-valued r.e.’s.

Let q be a (j£f, ^-measurable p-subhomogeneous (0 < p ^ 1) seminorm defined on a measurable vector space (L, f£). Furthermore, let {X„} be a sequence of r.e.’s taking values in L and let

We assume, as

j

previously, that {en} is a nondecreasing sequence of positive real numbers such that e„ -> oo.

To derive criteria (I), (II) and (III) we suppose that considered r.e.’s are independent.

An application of Kolmogorov’s three series theorem leads to the following result.

(I) If there exists a constant c > 0 such that

(О х р № . ) > « г ] < ® .

(ü)

and (hi)

Z E(eüpq(Xn))c < oo

Z a2(enpq(Xn))c < oo,

where (e„ pq{Xn))c = en pq{XJ) J [q {X „) ^ cep], then q(e„ l S„)-+ 0 a.s.

Let {mn} be a sequence of positive real numbers and let {gn} be a sequence of functions satisfying conditions: gn: R+ ->R + , gn(0) = 0, gn are continuous and nondecreasing, g„(x) ^ C0 for x ^ mn, gn{x) ^ C^x for 0 < x < m „ and gn(x)

^ C2x2 for 0 < x ^ mn.

(II) If ^Ед„(ейрд{Х„)) < oo, then q (e fl Sn) ^ 0 a.s.

(III) / / £ } P[q(Xn) $5 xeZ]dgn(x) <

, then q{e„ 0 a.s.

n 0

Proof. In view of Lemma 3.3 it suffices to show that pq(Xn) < со П

with probability 1.

Let q(Xn) = q{Xn)J lq {X n) < It can be easily verified that

I P [ ? w S’ m„en ^ C ô ‘ Z E g,(en P9(XJ) < со and

(4.1) C ^ P M X J S’ m„en+YlEg„(e-‘'q(X„))

n n

mn

< Z J Plq(Xn) > xep]dgn(x) < со.

n 0

Hence it follows that P[q(Xn) Ф q{Xn) i.o.] = 0. Moreover, Y ,e-pEq(X„) ^ C ^ Y ,E gn{e~i’ q(Xn)) < со and also

mn

Y,enpEq(Xn) « СГ11 J Plq(Xn) > xeUdg.ix) < со.

n n О

Therefore it remains to prove that the series

n

converges with probability 1. However,

Z<

2(e~pq(Xa)) « C J1 l 1Eg„(e-”q(Xn)) < со,

n n

which is a consequence of the obvious a.s. inequality q(Xn) < q(Xn) and (4.1). • This completes the proof.

The above results are quite satisfactory and e.g. the Marcinkie- wicz-Zygmund [3] law of large numbers for independent identically distrib­

uted summands follows easily as a corollary from (II) or (III). But as we shall see later, the first part of the mentioned theorem can be proved under less restrictive assumptions, therefore we shall derive besides two other criteria for dependent r.e.’s.

Let F be a class of real functions f : R+ ->R + which are continuous, nondecreasing and such that /(0 ) = 0, f(t) > 0 for t > 0 and their quotients t/f{t) are nondecreasing as t -> oo.

(IV) / / / „

F and Z ^ /n(4(*„))//>«) < » , then q{e^1Sn) ^ 0 a.s.

Proof. We argue similarly as Woyczynski [5] proving his Theorem 5.

Setting f ( t ) = min(t, 1) for t ^ 0, we see that for x eL , /п(я(*))//и( 0 ^ fn(ep nq(x/en))/fn(ep n) >f(q{x/en)).

Thus it suffices to show that the convergence of £E/(g(X„/e„)) implies our statement.

m

On account of subadditivity of/w e have £ / ( £ q(Xi/el))-^ 0 as n, m-> oo.

i — n

Let Lf = { r - Q->R; Ef(\q>\) < oo} be a metric space equipped with the distance d{cp, ф) = Е/(\ф — ф\). Since Lf is then complete, £g(X„/e„) converges

П

in Lf , and consequently Mn = £ qiXJe^-^O in Lf . By Fatou-Lebesgue n

theorem liminf/(M„) = 0 a.s., and hence Mn-+ 0 with probability 1 because n

( M j is monotone. Therefore the series £ q (X n/e„) converges a.s. and an П

application of Lemma 3.3 gives the desired conclusion.

We also have a criterion for r.e.’s with a nonintegrable seminorm, which is assumed now to be symmetric.

Let f neF and let ( X j be a sequence of r.e.’s in L not necessarily independent.

(V) I fY 1Efn(q(XJ)/[fn(q(Xlt))+fn(e^ )]< co, then q (e ^ S J ^ 0 a.s.

n

Proof. Let Z„ = X nS[q(X% < ep]. Then

Hence (4.2)

E f M Z M ( e0 + p № . ) » O « 2E /.(«(* .)) /„(«(*„))+/.(<*)'

l£ /„(« (Z „))//„(0 < ® and

(4.3) Y.PlZn* X J < c o .

On the basis of (4.2) and (IV), q(en a.s., and in view of (4.3), Borel-Cantelli lemma and linear continuity of q at zero, q(e„ 1Sn)->0 a.s. j

5. Marcinkiewicz-Zygmund type law of large numbers in a measurable

vector space. As an example of an application of the above results we shall

prove the strong law of large numbers of Marcinkiewicz-Zygmund [3] type for

a sequence of identically distributed (i.d.) r.e.’s in a measurable vector space L.

T

5.1. Let q be a symmetric seminorm and let {X n} be a sequence of i.d.r.e.’s in L.

(a) Let f

F and {e„} be such that

X 1 //(ef) = 0(n/f(e')).

л

If q is p-subhomogeneous and ^ P [q (X i) e£] < oo, then q(e~*Sn)-+0 a.s.

П

(b) If X n are independent, q is p-homogeneous and </(en ~1 Sn) —» 0 a.s., then ХР[«(-Х,) > еЯ < oo.

Л

Proof, (a) Let Z„ be defined as in (V). It is easy to see that en

X Ef(q{ZJ)lf(ef) = Х т / й \ f W n q W ,) < *]

n n J \€n) J

0

<

X

- , « «(X,) < <*]

n J Уеп)Шп

= 1 1 Ш / М - ^ д ( Х 1)<еП

i n ^ i J Ус п)

^ const•Yii'P [ef-1 ^ qiXJ < ef]

I

= const-X X p l>f-i < < ef]

i tt ^ i

= const X p [ef-i ^ 9 (^ 1 ) < ef]

л л

= const-^ P lqiX j) ^ e j-i] < 00 . Л

Hence, according to (IV), we have q(e„ 1 £ Z7) -» 0 a.s. The rest of the proof goes

j

along the lines of that of (V).

(b) Since q is nondecreasing, q(e~1Sn- l) ^ q(eüfl S „-1)-^0 a.s., therefore q ie^ X J < q{eü1SH ) + q( — e~1S„_1)->0 a.s. The last relation means that P[q(Xn) ^ e% i.o.] = 0, which together with the Borel zero-one law entails assertion (b).

Theorem 5.1 combined with an inequality connecting moments and series

of tails of r.v.’s allows us to establish the Marcinkiewicz-Zygmund law of large

numbers for r.e.’s in a measurable vector space L. We quote below the

mentioned necessary result stated as lemma.

Le m m a

5.2. For an arbitrary Щ-measurable function q: L-+R+ and real numbers s, x > 0 we have

x^ P L q iX ) ^ n1/sx] ^ Eqs(X) ^ xs £ P lq(X) >

1/

],

n

where X is a r.e. in L.

Proof. See Loéve [2], p. 242, the moments lemma.

Co r o l l a r y

5.3. Let q and

satisfy the same assumptions as in Theorem 5.1 ((a) and (b)), respectively), and let en — nl/r, where 0 < r < p.

(a) If Eqrlp(X 1) < oo, then q(n~1/rSn)-^0 a.s.

(b) If q(n~1,rS„)-+ 0 a.s., then Eqrlp{X l) < oo.

Proof. L et/(x) = x. Obviously, / e F and 00

£ 1 /f(ef) ~ const- J x~p,rdx = 0{n/ep).

i ^ n n

Our result is now a simple consequence of Theorem 5.1 and Lemma 5.2.

6. Marcinkiewicz-Zygmund type law of large numbers in linear metric spaces. Let L be a linear metric space equipped with distance d and let x be a topology in L generated by d. The definition of the linear metric space postulates continuity of operations + : (x, j/)->x + y and •: (a, x)->-ax from topological product spaces L ® L and R ® L respectively, into L. The con­

tinuity of addition and multiplication by scalars implies that shifts and homotheties of open sets in L are still open, i.e.,

Д Д Д (G +

)

and cGex.

Get x e L с Ф 0

Although translations do not change the topology i, a linear metric space L with its Borel tr-field & (L) need not be a measurable vector space, because in general addition is $ (L (g> L)-measurable, where the Borel G-field of topological product

contains ^(L)x.^(L) and the inclusion may be proper.

Therefore to make possible an adaptation of our previous results to L throughout this section we must and do assume that f£ is equal to the smallest cr-field with respect to which all the considered seminorms are measurable and (L, f£) is a measurable vector space. We may take into consideration J'(L), but then we have to suppose that operations + and ■ are J'(L) x @i(L) and x ^(L)-measurable, resp. Such an additional assumption is unnecessary provided L is separable, or equivalently L satisfies the second axiom of countability (see, e.g. Billingsley [1], Appendices (I) and (II)).

Obviously, F*-spaces, countably normed and normed spaces (including

Fréchet, B0, Banach and Hilbert spaces) are linear metric spaces, so that they

may be used as L provided the above mentioned additional conditions are

satisfied. As an example of a nonseparable linear metric space which is a measurable vector space may serve the space /°° of bounded sequences x = {£„}, endowed with the norm ||x|| = sup|£J.

П

Recall that a set A cz L is said to be starlike if aA cz A for all 0 ^ a < 1.

The modulus of concavity of a starlike set A is defined as a number e(A) = inf{a > 0 : A + Acz a A}

oo

if A + A cz aA for some a > 0, otherwise.

A starlike set A with a finite modulus of concavity is called pseudoconvex, and a linear metric space L possessing a basis У of neighbourhoods of zero consisted of pseudoconvex sets is termed as a locally pseudoconvex space.

It is well known (cf. Rolewicz [4], Chapter III) that if L is a locally pseudoconvex linear metric space, then there exists a sequence {q f of prhomogeneous symmetric seminorms determining a topology equivalent to the original one. If, moreover, c(A) ^ 2 1/p for all AeY', 0 < p ^ 1, then we can take Pi = p for i ^ 1 and in such a case L is said to be locally p-convex.

Theorem 5.1 may be rewritten now as follows:

Theorem 6.1.

Let L be a locally pseudoconvex linear metric space with a topology generated by a sequence {qt, i ^ 1} of Pfhomogeneous seminorms and let {X„} be a sequence of i.d. r.e.’s in L.

(a) Let fi&F and {en} be such that for each i ^ 1, X 1 lf{e f) = 0(n/f(ep%

j>n

If £ PLqfXi) ^ ePi~\ < oo for all i ^ 1, then e f 1Sn-+ 0 a.s. in L.

П

(b) If X n are independent and e f 1Sn^-0 a.s. in L, then

Ï=ï 1 n

Obviously, in an analogous way one can transform criteria (I)-(V) and Corollary 5.3 and adapt them to locally pseudoconvex linear metric spaces, but in the case of Corollary 5.3 we have to assume that p0 — infpt > 0.

i

The presented approach enable us to obtain also similar laws of large numbers for a wide class of modular spaces.

A function

defined on a linear space

is called a p-convex modular, 0 < p ^ l , if @(x) = 0<=>x = 0, ^(x) = ^( —x) and Q(<xx + fy) ^ %

{

) + fipQ{y) for a, f ^ 0, otp + pp = 1. Setting Lg = { x e f : p(Àx)

< oo for some À > 0} we observe that Le is a linear space. Lg equipped with a topology

determined by the convergence relation

x„-*0<=>£(Axn)->0 for each Д > 0

is said to be a p-convex modular space.

Let q(x) = inf{a > 0: p(x/a1/p) ^ 1}. Then q: Le ->Rt is a p-homo- geneous symmetric seminorm in Le such that q(x) = 0<=>x = 0 and q{xn) - + 0 o x n->0 in Lq, i.e., q generates a topology equivalent to the original one (note that q satisfies the triangle inequality since it is p-homogeneous).

It is clear that criteria (I)-(V), Theorem 5.1 and Corollary 5.3 are then valid for sequences of r.e.’s taking values in a modular space Lg.

In particular, the last remark relates to Orlicz spaces determined by a function Ф: R + - > R + such that (а) Ф is continuous and nondecreasing, (b) Ф(0) = 0, Ф(г) > 0 for t > 0 and Ф(0->оо as t-*co, and, moreover, (c) Ф is p-convex, i.e., Ф(cct1 + pt2) ^ арФ((1) + ^рФ((2) for a, /? ^ 0, ap + /?p = 1, tl , t 2e R +. The simplest examples of such Ф-functions are Ф1(г) = tp and ф2(0

Ф(1Р), where W: R + R + is a convex function.

More generally, let sé be a «7-field of subsets of a set A and let p be a measure on a measurable space {A, sé). Assume that Ф: A x R + -^R+ is for each fixed t^ O an ^/-measurable mapping from A into R + and for p-a.a.

и

А, Ф(и, •) is Ф-function having properties (a), (b), (c) as above. Then

q

(

) = j Ф(и, \x(u)\)n(du)

A

is a p-convex modular on a set 3C of p-equivalence classes of ^/-measurable real functions on the space A and the generalized Orlicz space 1? = Le generated by g satisfies (I)-(V), Theorem 5.1 and Corollary 5.3.

By analogy to the previous considerations we can also generalize the last statements to modular spaces generated by a sequence {pt} of prconvex semimodulars, such that x = 0o^ .(x) = 0 for all i ^ 1. Taking L— f ] L ei, we

i

define q( as above by means of and formulate Corollary 5.3 (if p0 > 0) and (IHV) similarly as Theorem 6.1. This procedure is available in particular for generalized Orlicz spaces induced by locally prconvex Ф-functions, i.e., Ф-functions satisfying the following condition: there exists a denumerable partition {TJ of A on disjoint ^/-measurable sets A{ such that for p-a.a. u e A t, Ф(и, •) is a pr convex Ф-function. The generalized Orlicz space determined by the family of semimodulars

Qi{x) = f Ф(и, \x(u)\)n{du) Ai

can be used then instead of L in Theorem 6.1 and extended criteria (I)-(V) or Corollary 5.3 (with p0 > 0).

Acknowledgement. The author is grateful to Professor D. Szynal for his

helpful remarks and advices obtained during the writing of this paper.

References

[1 ] P. B i l l i n g s l e y , Convergence o f Probability Measures, W iley, New York 1968.

[2 ] M . L o é v e , Probability Theory, 2nd ed., Van Nostrand, New York 1960.

[3 ] J. M a r c i n k i e w i c z and A . Z y g m u n d , Sur les fonctions indépendantes, Fund. M ath. 29 (1937), 6 0 - 9 0 .

[4 ] S. R o l e w i c z , M etric Linear Spaces, P W N , Warsaw 1972.

[5 ] W . A. W o y c z y r i s k i , Strong laws o f large numbers in certain linear spaces, Ann. Inst. Fourier, Grenoble 24 (1974), 2 0 5 -2 2 3 .

IN STITU TE O F M A T H E M A TIC S

M A R IA C U R IE -S K L O D O W S K A U NIVERSITY LU BLIN