Discrete limit theorems for general Dirichlet series. III

Transcription

1 CEJM iscrete limit theorems for general irichlet series. III A. Laurinčikas a,r.macaitienė epartment of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT Vilnius, Lithuania Received 24 February 2004; accepted 23 April 2004 Abstract: Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general irichlet series. The explicit form of the limit measure in this theorem is given. c Central European Science Journals. All rights reserved. Keywords: irichlet series, probability measure, random element, weak convergence MSC 2000: M4, 30B50, 60B0 a Partially supported by Lithuanian Foundation of Studies and Science Introduction Let s σ + it be a complex variable, and let R and C denote the set of real and complex numbers, respectively. The series a m e λms, where a m C and λ m R, 0<λ <λ 2 <..., lim m λ m +, is called a general irichlet series with coefficients a m and exponents λ m. If λ m logm, thenwehavethe ordinary irichlet series a m m. s antanas.laurincikas@maf.vu.lt renata.macaitiene@mif.vu.lt

2 340 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Suppose that absolutely converges for σ>σ a and has a sum fs. Then the function fs is regular in the half-plane σ>σ a. The value-distribution of the function fs is complicated, therefore probabilistic methods were used to begin attacking the problem. It is convenient to state probabilistic results for functions given by irichlet series in the form of limit theorems in the sense of the weak convergence of probability measures. This idea belongs to H. Bohr who jointly B. Jessen [2], [3] obtained limit theorems for the Riemann zeta-function. Later many mathematicians A. Wintner, V. Borchsenius, A. Selberg, A. Ghosh, P..T.A. Elliott, E. Stankus,. Joyner,. Hejhal, E. M. Nikishin, B. Bagchi, K. Matsumoto, R. Garunkštis,W.Schwarz,J.Steuding,R.Šleževičienė, R. Kačinskaitė, J. Ignatavičiūtė, I. Belov, the authors and others extended and generalized Bohr Jessen s results. In general, the probabilistic value-distribution of the ordinary irichlet series has been widly studied, while results of such a kind for general irichlet series are not numerous. enote by meas{a} the Lebesgue measure of the set A R, and let νt t... meas{t [0; T ]:...}, where in place of dots, a condition satisfied by t is written. In T [4] the distribution function νt t F t <x, where F t Rfσ+itorF t Ifσ+it, was considered in connection with the Besicovitch classes. Let G be a region on the complex plane C. enote by HG the space of analytic functions on G equipped with the topology of uniform convergence on compacta, and let BS stand for the class of Borel sets of the space S. The paper [6] is devoted to weak convergence of the probability measure ν τ T fs + iτ A, A BHa, where a {s C : σ>σ a }. It was proved that the latter measure converges weakly to some probability measure on H a, BH a as T, while in the case of a linear independence of the system {λ m } over the field of rational numbers the explicit form of the limit measure was given. The further investigations in this field are related to limit theorems in the space of meromorphic functions. enote by MG the space of meromorphic functions on G equipped with the topology of uniform convergence on compacta. Suppose that the function fs is meromorphically continuable to the region σ>σ with some σ <σ a, all poles in this region being included in a compact set. Moreover, we assume that, for σ>σ, the estimates and t 0 is a fixed positive number, and fσ + it O t α, t t 0,α>0, 2 2T T T fσ + it 2 dt OT, T, 3 are satisfied. Let {s C : σ>σ }. Then in [7] it was proved that the probability measure ν τ T fs + iτ A, A BM,

3 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics weakly converges to some probability measure P on M, BM, however, the explicit form of this measure was not indicated. The first attempt to find the limit measure P in the theorem of [7] was made in [9]. Assuming additionally that λ m clog m δ, 4 where c and δ are some positive constants, and that the set {log 2} {λ m } 5 is linearly independent over the field of rational numbers; in [9] the explicit form the limit measure P was obtained. Finally, in [4] the number log 2 from 5 was removed, so the explicit form of the measure P is known if the system of exponents {λ m } is linearly independent over the field of rational numbers. In [8] the weak convergence for σ>σ of the probability measure ν t T fσ + it A, A BC, was considered as T. All mentioned above limit theorems for the function fs have a continuous character: in these theorems the studied measures are defined by translations fσ + it orfs + iτ, where t and τ vary continuously in the interval [0; T ]. An another discrete statement of the problem is also possible. In this case t or τ takes values in some arithmetical progression. In [0] we began the investigation of the discrete value-distribution of by probabilistic methods, and we proved for it limit theorems in the sense of the weak convergence of probability measures on the complex plane. iscrete limit theorems for fs inthe space of analytic functions were obtained in [2]. Let µ N... #{0 m N :...}, N + where in place of dots a condition satisfied by m is written. Let γ denote the unit circle on C, i.e. γ {s C : s }, andlet Ω γ m, where γ m γ for all m N. With the product topology and pointwise multiplication, the infinite-dimensional torus Ω is a compact topological Abelian group proof can be found in [5]. Therefore, there exists the probability Haar measure m H on Ω, BΩ, and this gives a probability space Ω, BΩ,m H. Let ωm stand for the projection of ω Ω to the coordinate space γ m. Suppose, as above, that the function fs is meromorphically continuable to the halfplane σ>σ, σ <σ a, and that all poles in this region belong to a compact set. Moreover, we require that, for σ>σ, the estimates 2 and 3 should be is satisfied.

4 342 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Now, for σ>σ, on the probability space Ω, BΩ,m H we define a complex-valued random element fσ, ω by fσ, ω a m ωme λmσ, and denote by Q f the distribution of the random element fσ, ω. In this paper we suppose that h>0 is fixed and such that exp{ 2π } is a rational number. Then in [0] the h following theorem was proved. Theorem.. Suppose that the function fs satisfies conditions 2 and 3, {λ m } is a sequence of algebraic numbers linearly independent over the field of rational numbers, and satisfies condition 4. Then the probability measure Q N A µ N fs + imh A, A BC, converges weakly to Q f as N. Now define an H a -valued random element fs, ω onω, BΩ,m H by fs, ω a m ωme λms, s a, and denote by Q f its distribution. Then in [2] the following theorem was obtained. Theorem.2. Suppose that {λ m } is a sequence of algebraic numbers linearly independent over the field of rational numbers. Then the probability measure converges weakly to Q f as N. Q N A µ N fs + imh A, A BH a, The aim of this paper is to obtain a discrete limit theorem for the function fs in the space of meromorphic functions. Let C C { } be the Riemann sphere, and let ds,s 2 be a metric given by the formulae ds,s 2 2 s s 2 + s 2 + s 2, ds, 2, d, 0, 2 + s 2 where s, s,s 2 C. This metric is compatible with the topology of C. enote by MG the space of meromorphic functions g : G C,d equipped with the topology of uniform convergence on compacta. In this topology, a sequence {g n : g n MG} converges to the function g MG if dg n s,gs 0, n,

5 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics uniformly on compact subsets of G. Consider the weak convergence of the probability measure P N A µ N fs + imh A, A BM. On the probability space Ω, BΩ,m H defineanh-valued random element fs, ω by the formula fs, ω a m ωme λms, s. 6 enote by P f the distribution of the random element fs, ω. Then we have the following statement. Theorem.3. Suppose that {λ m } is a system of algebraic numbers linearly independent over the field of rational numbers and satisfies condition 4, and that the function fs satisfies conditions 2 and 3. Then the probability measure P N converges weakly to P f as N. 2 Limit theorems for irichlet polynomials Since all poles of the function fs in the region are included in compact set, the number of these poles is finite. enote them by s,...,s r,andlet f s r e λ s j s. j Then, clearly, f is a irichlet polynomial, and f s j 0 for j,...,r.moreover,let f 2 s f sfs. Then we have, that the function f 2 s is regular on. enote by A the number of elements of the set A. Then, for σ>σ a,wehave f 2 s r e λ s j s a m e λms j A {,...,r} j0 a m e λ s j A e λm+ A λ s a mj e λm+jλs, with some coefficients a mj satisfying a mj O a m for m N and j 0,,...,r.Here the first sum runs over all the subsets A of {,...,r}. Obviously, the definition of f 2 s and conditions 2 and 3 imply, for σ>σ,the estimates f 2 s O t α, t t 0,α>0, 7

6 344 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics and T T f2 σ + it 2 dt OT, T. 8 Let S and S 2 be two metric spaces. For us the following statement will be useful. Lemma 2.. Let P, P n be probability measures on S, BS, and h : S S 2 be a continuous function. Then the weak convergence of P n to P implies the weak convergence of P n h to Ph on S 2, BS 2 as n. The lemma is a particular case of Theorem 5. from []. We begin with a limit theorem for the irichlet polynomial p n s n a mj e λm+jλs. j0 Lemma 2.2. Suppose that {λ m } is a sequence of algebraic numbers linearly independent over the field of rational numbers. Then there exists a probability measure P pn on H, BH such that the probability measure P N,pn A µ N pn s + imh A, A BH, converges weakly to P pn as N. Proof. Let n Ω n γ m, where γ m γ for all m,...,n. efine the function u :Ω n H by the formula ux,...,x n n j0 a mj e λm+jλ s x j x m, x,...,x n Ω n. Then we have that u is a continuous function, and p n s + imh ue iλ mh,...,e iλnmh. 9 Consider the probability measure P N on Ω n, BΩ n defined by P NA µ N e iλ mh,...,e iλnmh A. The Fourier transform g N k,...,k n, k,...,k n Z, of the measure P N is g N k,...,k n Ω x ik...x ikn n dp N N + N e imh n k l λ l l.

7 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Since {λ m } is a system of algebraic numbers linearly independent over the field of rational numbers and h has the above properties, we find [] that ifk,...,k n 0,...,0, g N k,...,k n exp{in+h n k l λ l } l ifk N+,...,k n 0,...,0. Consequently, lim g Nk,...,k n exp{ih n k l λ l } l { ifk,...,k n 0,...,0, 0 ifk,...,k n 0,...,0. By Theorem.3.9 of [5] this implies that the measure P N converges weakly to the Haar measure m nh on Ω n, BΩ n as N. Hence, the continuity of the function u, Eq. 9, and Lemma 2. yield that the probability measure P N,pn converges weakly to P pn m nh u as N. Now let gm, m N, be complex numbers such that gm for all m N. efine p n s, g n a mj gme λm+jλs. j0 Lemma 2.3. The probability measure P N,pn A µ T pn s + imh, g A, A BH, also converges weakly to the measure m nh u as N. Proof. Let θ m arggm, m 0,,..., n, and let the function u : Ω n Ω n be given by the formula Then we have that u x,...,x n x e iθ,...,x n e iθn, x,...,x n Ω n. p n s + imh, g u u e iλ mh,..., e iλnmh. Therefore, by the proof of Lemma 2.2, the measure P T,pn converges weakly to the measure m nh uu as N. Since the Haar measure is invariant with respect to translations by points from Ω n, P N,pn converges weakly to m nh u u m nh u as N. 3 Approximation in the mean In this section we approximate the function f 2 s, defined at the beginning of Section 2, by an absolutely convergent irichlet series in the mean. Let σ 2 σ a σ.forσ [ σ 2,σ 2 ] define l n s s s Γ e λn+jλs, σ 2 σ 2

8 346 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics where, as usual, Γs is the gamma-function. Obviously, σ 2 > 0. For σ>σ consider the function g n s σ2 +i f 2 s + zl n z dz 2πi z. σ 2 i Lemma 3.. The function g n s has the expansion g n s a mj exp { e } λm λnσ 2 e λm+jλs, 0 j0 the series being absolutely convergent for σ>σ. Proof. Since σ 2 σ a σ, it follows that σ +σ 2 >σ a for σ>σ. Therefore, for Rz σ 2, We put and consider the series f 2 s + z k n m 2πi j0 j0 σ2 +i σ 2 i a mj e λm+jλs+z. l n se λm+jλ s ds s, a mj k n me λm+jλs+z. Since k n m O e λm+jλ σ 2 ln σ 2 + it dt O e λm+jλ σ 2, the series converges absolutely for σ>σ a σ 2, i. e., for σ>σ. Hence we may interchange summation and integration in the definition of g n s. This gives g n s j0 j0 a mj e λm+jλ s 2πi σ2 +i σ 2 i l n ze λm+jλ z dz z a mj k n me λm+jλs. 2 Using the equality 2πi c+i c i Γsb s ds e b, c > 0, b>0, we find that 2πi σ2 +i σ 2 i σ2 +i s s λm λns ds Γ e 2πi σ 2 i σ 2 σ 2 s s σ Γ e λm λn s σ 2 2 d s exp { e } λm λnσ 2. σ 2 σ 2

9 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Hence and from Eq. 2 the lemma follows. Lemma 3.2. Let T 0 and T δ>0berealnumbers,t be a finite set in the interval [T 0 + δ,t T δ ]. Moreover, let 2 N δ x t T t x <δ and let Sx be a complex-valued continuous function on [T 0,T+ T 0 ] having a continuous derivative on T 0,T + T 0. Then, t T N δ t St 2 δ + T 0 +T T 0 T 0 +T Sx 2 dx T 0 +T Sx 2 2 dx S x 2 2 dx. T 0 T 0 Proof. This is Lemma.4 of [3]. Lemma 3.3. Let T. Then, for σ>σ, T 0 f 2σ + it 2 OT. Proof. By the Cauchy formula f 2s 2πi z s δ f 2 z z s 2 dz, where the circle z s δ lies in the half-plane σ>σ. Then for some σ >σ and bounded τ by Eq. 8 T 0 f 2σ + it 2 dt T 0 2πi z s δ f 2 z z s dz 2 T dt O f2 σ + it + iτ 2 dt OT. 0 Lemma 3.4. Let K be a compact subset of. Then lim lim sup n N + N sup f 2 s + imh g n s + imh 0. s K Proof. We change the contour in the integral for g n s. The integrand has a simple pole at the point z 0. Suppose σ [σ + η, σ 4 ],η>0, when s K, and put σ 3 σ + η 2.

10 348 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Then, by the residue theorem, we find g n s 2πi σ 3 σ+i σ 3 σ i f 2 s + zl n z dz z + f 2s. 3 Now let L be a simple closed contour lying in and enclosing the set K, andletδ stand for the distance of L from the set K. Then by the Cauchy formula, for s K, wehave f 2 s + imh g n s + imh f 2 z + imh g n z + imh dz. 2πi z s Therefore, sup f 2 s + imh g n s + imh s K 2πδ L L f 2 z + imh g n z + imh dz. Hence, for sufficiently large N, weobtain N sup f 2 s + imh g n s + imh N + O N O N N L O Nδ s K 2πδ dz L 2N + O N sup σ Remembering Eq. 3, we have f 2 z + imh g n z + imh dz f 2 Rz + imh g n Rz + imh + O Nδ 2N s L f 2 s + imh g n s + imh O f 2 σ + imh g n σ + imh. 4 f 2 σ 3 + imh + iτ l n σ 3 σ + iτ dτ. Hence, taking u [ τ ] +, where [x] denotes the integer part of x, we obtain that h N 2N f 2 σ + imh g n σ + imh O By Lemmas 3.2 and 3.3, and estimate 8 2N+u m u f 2 σ + imh 2 h + 2N+uh uh 2N+uh f 2 σ + it 2 dt f 2 σ + it 2 dt l n σ 3 σ + iτ N 2N+uh 2N+u m u f 2σ + it 2 dt f 2 σ 3 + imh dτ. 2 O2N +2u. uh uh

11 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Therefore the Cauchy - Schwarz inequality yields N + sup σ O 2N s L sup σ s L O sup σ s L O sup σ s L f 2 σ + imh g n σ + imh 2N+u l n σ 3 σ + iτ dτ f 2 σ 3 + imh 2 2 N + m u 2N +2u l n σ 3 σ + iτ N + dτ l n σ 3 σ + iτ + τ dτ. 5 We can choose the number δ so that the inequality σ 3 σ η, s L, should be 4 satisfied. In this case by definition of l n s, we have that lim sup n σ η 4 l n σ + it + t dt 0. The above along with estimates 4 and 5 prove the lemma. Let a h {e iλmh : m N}, h>0. Then a h is a one-parameter group. We define the one-parameter family ϕ h of transformations on Ω by ϕ h ω a h ω for ω Ω. Then ϕ h is a measurable measure preserving transformation on the probability space Ω, BΩ,m H. Now we recall some basic facts of ergodic theory, see for example [5]. Let G be a compact topological Abelian group with Haar measure m G.Let{G h } be an one-parameter group of measurable transformations on G. A set A BG is called an invariant set with respect to the group {G h } if, for each h, thesetsa and A h G h A differ from one another by a set of zero m G -measure, i.e. m G A A h 0, where A A h denotes the symmetric difference of sets A and A h. A one-parameter group {G h } is called ergodic if its σ-field of invariant sets consists only of sets having m G -measure equal to 0 or. For these and other facts of ergodic theory, see for example, [5]. Lemma 3.5. The one-parameter group ϕ h is ergodic. Proof. This is Lemma 7 from [0]. Let, for s, f 2 s, ω j0 a m ω j ωme λm+jλs. Then we have that f 2 s, ω is a product of two H-valued random elements, r f s, ω ωe λ s j s j

12 350 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics and fs, ω, and therefore it is an H-valued random element. enote by Eξ the mean of the random element ξ. Lemma 3.6. Let T be a measurable measure preserving ergodic transformation on probability space Ω, F,m. Then for every g L Ω, F,m n lim gt k ωeg n n for almost all ω Ω. k0 The lemma is the well-known Birkhoff-Khinchine theorem, see for example [6]. Lemma 3.7. Let σ>σ and N.Then for almost all ω Ω. Proof. Let, for m N, and let, for a fixed j, N f 2 σ + imh, ω 2 BN f mj σ, ω a mj ω j ωme λm+jλ σ, f j σ, ω f mj σ, ω. Then we have Ef mj, f kj a mj a kj e λm+jλσ e λ k+jλ σ since { amj 2 e 2λm+jλσ, m k, 0, m k, Ω {, m k, ωmωkm H dω 0, m k. Ω ω j ω j ωmωkm H dω Thus we have that the random variables f mj σ, ω are orthogonal. In [9] it was proved that, for σ>σ, a m 2 e 2λmσ <. Consequently, E f j σ, ω 2 E f m,j σ, ω 2 O a m 2 e 2λm+jλ σ <, j 0,.., r.

13 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Hence, Clearly, E f 2 σ, ω 2 E f j σ, ω 2 <. 6 j0 f 2 σ, ϕ m h ω 2 f 2 σ, a mh ω 2 f 2 σ + imh, ω 2. 7 Therefore, in view of 6 and Lemmas 3.5 and 3.6, we find lim N + N f 2 σ + imh, ω 2 lim for almost all ω Ω. Hence the lemma follows. Let, for ω Ω, N + N E f 2 σ, ω 2 < f 2 σ, ϕ m h ω 2 g n s, ω a mj ω j ωmexp{ e λm λnσ 2 }e λm+jλs. j Clearly, the last series converges absolutely for σ>σ. Lemma 3.8. Let K be a compact subset of. Then for almost all ω Ω. lim lim sup n N + N sup f 2 s + imh, ω g n s + imh, ω 0 s K Proof. In virtue of Lemma 3.7, the proof is similar to that of Lemma Limit theorems for g n s In this section we consider the weak convergence of two probability measures on H, BH, namely, P N,n A µ N gn s + imh A, and P N,n A µ N gn s + imh, ω A as N. To investigate the weak convergence of these measures we need a metric on H which induces its topology. It is known, see, for example, Lemma.7. of [5], that there exists a sequence {K n } of compact subsets of such that nk n, K n K n+,andifk is a compact subset of, thenk K n for some n. Then ϱf,g n ϱ n f,g, f,g H, +ϱ n f,g

14 352 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics where ϱ n f,g sup fs gs s K n is a metric on H which induces its topology. Lemma 4.. There exists a probability measure P n on H, BH such that both the measures P N,n and P N,n converge weakly to P n as N. Proof. efine, for a positive integer M, g n,m s r M j0 a mj exp { } e λm λnσ 2 e λ m+jλ s, g n,m s, ω M a mjω j ωmexp { e 2} λm λnσ e λ m+jλ s, and let P N,n,M A µ N gn,m s + imh A, A BH, P N,n,M A µ N gn,m s + imh, ω A, A BH. By Lemmas 2.2 and 2.3, both the measures P N,n,M and P N,n,M converge weakly to the same measure P n,m as N. Similarly as in Theorem of [5], we obtain that the family of probability measures {P n,m } is tight for fixed n. Hence, by the Prokhorov theorem, see for example, Theorem..2 of [5], it is relatively compact. By the definition of g n s andg n,m s, we have lim M g n,ms g n s, and since the series for g n s converges absolutely for σ>σ, the convergence is uniform on compact subsets of. Hence, for every ε>0, lim n lim sup µ N ϱgn,m s + imh,g n s + imh ε lim M lim sup N+ɛ N ϱ g n,m s + imh,g n s + imh 0. 8 Let θ N be a random variable on a certain probability space Ω, B Ω, P with distribution We put enote by Pθ N mh, m 0,,..., N. N + X N,n,M s g n,m s + iθ N. the convergence in distribution. Then by the above remark X N,n,M s X n,m, where X n,m is an H-valued random element with the distribution P n,m.nowlet X N,n s g n s + iθ N.

15 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Then, in view of Eq. 8, for every ε>0 lim lim sup P ϱx N,n,M s,x N,n s ε 0. 9 Since the family {P n,m } is relatively compact, there exists a subsequence P n,m converges weakly to P n,say,asm.then which X n,m M P n. 20 The space H is separable. Therefore, by relations 8 20 and Theorem.2.4 of [5], X N,n P n. 2 This means that there is a probability measure P n such that P N,n converges weakly to P n as N. On the other hand, relation 2 shows that the measure P n is independent of the choice of the subsequence P n,m. This and the relative compactness of {P n,m } imply the weak convergence of P n,m to P n as M, and also the relation X n,m M P n. 22 Now repeating the same arguments for the random elements X N,n,M s, ω g n,m s + iθ, ω, X N,n s, ω g n s + iθ, ω, and taking into account relation 22, we obtain that the measure P N,n also converges weakly to P n as N. The lemma is proved. 5 Limit theorems for f 2 s In this section, we will consider the weak convergence of probability measures P N,f2 A µ N f2 s + imh A, P N,f2 A µ N f2 s + imh, ω A, A BH, A BH. Lemma 5.. There exists a probability measure P on H, BH such that both the measures P N,f2 and P N,f2 converge weakly to P as N. Proof. The way of the proof is the same as in Lemma 4.. By this lemma, the measures P N,n and P N,n converge weakly to the same measure P n as N.Taking X N,n s g n s + iθ N, we have that X N,n X n, 23

16 354 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics where X n is an H-valued random element with the distribution P n. Also, the family {P n } is relatively compact. Applying Lemma 3.4, we find that, for every ε>0, Now we set Then we can write lim n lim n lim sup lim sup µ N ϱf2 s + imh,g n s + imh ε N lim n N +ε ϱf 2 s + imh,g n s + imh 0. Y N s f 2 s + iθ N. lim sup P ϱx N,n s,y N s ε From the relative compactness of {P n } there exists a subsequence {P n } of {P n } which converges weakly to P as n, i.e., X n n P. 25 Using and applying Theorem.2.4 of [5] again, we obtain that Y N n This means that the measure P N,f2 converges weakly to P as N. The relative compactness of {P n } and relation 25 show that X n n P. 26 Repeating the above arguments, and using 26 and Lemma 3.8, we obtain that the measure P N,f2 also converges weakly to P as N. The aim of the next lemma is to identify the limit measure in Lemma 5.. As in Section 3, for s, wedefine f 2 s, ω a mj ω j ωme λm+jλs. j0 Let P f2 be the distribution of the random element f 2 s, ω. Lemma 5.2. The measure P in Lemma 5. coincides with P f2. Proof. Let A BH be a continuity set of P. Then by Lemma 5. lim µ N f2 s + imh, ω A P A 27 for almost all ω Ω. Now we fix the set A and define the random variable θ on Ω, BΩ by the formula { iff2 s, ω A, θω 0 iff 2 s, ω A.

17 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Then Eθ By Lemmas 3.5 and 3.6 it follows that Ω θdm H m H ω Ω: f2 s, ω A P f2 A <. 28 lim N + N θϕ m h ω Eθ 29 for almost all ω Ω. However, the definitions of θ and ϕ h give N + N θϕ m h ω µ N f2 s + imh, ω A. 30 From Eqs we obtain lim µ N f2 s + imh, ω A P f2 A for almost all ω Ω. From this and Eq. 27, we have that P A P f2 A for any continuity set A of P. Since all continuity sets constitute a determining class, hence P A P f2 A for all A BH, and the lemma is proved. 6 A limit theorem for f s First we observe that f s r e λ s j s j b m e λ ms is a irichlet polynomial with some coefficients b m and exponents mλ, and define the H-valued random element f s, ω by f s, ω r ωe λ s j s j We define the probability measure b m ω m e λms. P N,f A µ N f s + imh A, A BH. Lemma 6.. The probability measure P N,f converges weakly to the distribution P f of the random element f s, ω asn. Proof. The lemma is obtained in the same way as Lemmas 2.2 and 5.2.

18 356 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics A two-dimensional limit theorem In Sections 5 and 6 we proved discrete limit theorems for the functions f s andf 2 s in the space H. Now we will prove a joint discrete limit theorem for these functions. Let H 2 H H. enote by P f,f 2 the distribution of the H 2 -valued random element F s, ω f s, ω,f 2 s, ω, ω Ω,s, and let P N,f,f 2 A µ N f s + imh,f 2 s + imh A, A BH 2. Lemma 7.. The probability measure P N,f,f 2 converges weakly to P f,f 2 as N. For the proof of this lemma we need the following results. Lemma 7.2. The family of probability measures {P N,f,f 2 } is relatively compact. Proof. In Lemmas 5., 5.2 and 6. we proved that the probability measures P N,f and P N,f2 converge weakly to the measures P f and P f2, respectively, as N. Therefore, the family of probability measures {P N,fj }, j, 2, is relatively compact. The space H is a complete separable space. Hence, by the Prokhorov theorem the family {P N,fj }, j, 2 is tight. Thus we have that for every ɛ>0there exists a compact subset K j such that P N,fj H\K j < ɛ, j, Let θ N be a random variable defined in the proof of Lemma 4., and f,n s f s + iθ N, f 2,N s f 2 s + iθ N, F N s f,n s,f 2,N s. Then inequality 3 and the definition of P N,fj yield P ɛ f j,n H\K j <, j, Let K K K 2.ThenKis a compact subset of H 2. In virtue of 32 P N,f,f 2 H 2 \K P F N H 2 \K P 2 f j,n s H\K j j 2 P f j,n s H\K j <ɛ. 33 j Relation 33 shows that the family {P N,f,f 2 } is tight. By the Prokhorov theorem it is relatively compact. The lemma is proved.

19 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Let s,..., s l be arbitrary points on, and put σ min k l Rs k.then σ σ σ < 0, and we set {s C : σ> σ}. Moreover, let u jk, j, 2, k l, be arbitrary complex numbers, and define a function v : H 2 H by the formula vf,f 2 2 l u jk f j s k + s, s, f j H, j, j k Let W v s v f s,f 2 s. Lemma 7.3. The relation holds. W v s + iθ N v F s, ω Proof. By the the definition of the function v, W v s 2 l u jk f j s k + s j k for σ>σ. In the region σ>σ a the f sandf 2 s are presented by absolutely convergent irichlet series r f s e λ s j s b j e λ js and f 2 s This shows that for s j j0 r e λ s j s a m e λms j j0 a m,j e λm+jλs. W v s l l u k f s k + s+ u 2k f 2 s k + s k l k u k k b j e λ js k +s + j0 bj e λjs + l u 2k l k j0 k j0 u 2k a mj e λm+jλ s k +s j0 ã mjk e λm+jλs Z s+z 2 s,

20 358 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics where and bj { l u k bj e λ bj s k, j r,, bj 0, j>r, k { ã mjk â mj e λm+jλ amj s k, j r,, â mj 0, j > r. Z s is a irichlet polynomial, and Z 2 s is a linear combination of irichlet series satisfying conditions 7 and 8. Since the function f 2 s is regular in the half-plane σ>σ, the function Z 2 s isregularin. Since the set {λ m } is a system of algebraic numbers linearly independent over the field of rational numbers, and the exponents have a property of type 4, we have, that the probability measure µ N Wv s + imh A µ N Z s + imh+z 2 s + imh A, A BH, 35 converges weakly to the distribution of the H -valued random element W v s, ω bj ωe λjs + l u 2k j0 k j0 ã m,j,k ω j ωme λm+jλ s 36 as N. The proof of this is obtained in a similar way like, for example, for the function f 2 s. First we prove a limit theorem for the measure µ N Z s+ l u 2k k j0 M ã mjk e λm+jλs A, A BH. After this, applying an approximation of the function W v s in the mean, it remains only to use some elements of the ergodic theory to obtain the explicit form of the limit measure, and it turns out that this limit measure coincides with the distribution of the random element defined by Eq. 36. However, by the definition of v W v s, ω l u k k j0 b j ω j e λ js k +s + 2 j0 k l u 2k k j0 a m,j,k ω j ωme λm+jλ s k +s u jk f j s k + s, ω v f s, ω,f 2 s, ω. Therefore, measure 35 converges weakly to the distribution of the random element v f s, ω,f 2 s, ω as N. Proof of Lemmas 7.. By Lemma 7.2 it follows that there exists a sequence N such that the measure P N,f,f 2 converges weakly to some probability measure P on H 2, BH 2 as N.Let F s f s,f 2 s

21 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics be an H 2 -valued random element with distribution P. Then by the choice of N we have that F N F N. 37 The function v is continuous. Hence, by Lemma 2., we have that vf N N vf. By the definition of W v s W v s + iθ N N vf. 38 enoting F s f s,f 2 s, by Lemma 7.3 we have W v s + iθ N N vf. From this and relation 38 vf vf. 39 Now let v : H C be defined by the formula v f f0, f H. Then the function v is measurable, and 39 gives or By the definition of v we find v vf vf 0 v vf, vf 0. 2 l u jk f j s k,ω j k 2 l u jk f j s k 40 j k with arbitrary complex numbers u jk. The hyperplanes in the space R 4n generates a determining class []. Then, the hyperplanes also form a determining class in C 2n. Hence, and from 40, we see that C 2n -valued random elements f j s k,ωandf j s k, j, 2, k,..., l, have the same distribution. enote by K a compact subset of, andletϕ,ϕ 2 H. Moreover, let for every ɛ>0, G {g,g 2 H 2 : sup g j s ϕ j s ɛ, j, 2}. s K We choose a sequence {s k } to be dense in K. Moreover,let G l {g,g 2 H 2 : g j s k ϕ j s k ɛ, j, 2, k,..., l}.

22 360 A. Laurinčikas, R. Macaitienė / Central European Journal of Mathematics Then the above mentioned properties of the random elements f j s k,ωandf j s k,ω show that m H ω Ω: F s, ω Gl P F s G l. 4 Since the sequence {s k } is dense in K, wehaveg G 2... and G l G as l.if l in Eq. 4, we find m H ω Ω: F s, ω G P F s G. 42 The space H 2 is separable. Thus finite intersections of the spheres form a determining class []. Hence, Eq. 42 implies F F. This and relation 37 yield F N N F. 43 Consequently, the measure P N,f,f 2 converges weakly to the distribution of random element F f s, ω,f 2 s, ω as N. Since by Lemma 7.2 the family {P N,f,f 2 } is relatively compact and the random element F in 43 is independent of the choice of N, by Theorem..9 of [5] and Lemma 7.2 we obtain the assertion of the lemma. 8 Proof of the Theorem efine the function u: The metric d satisfies H 2 M by the formula ug,g 2 g 2 g, g,g 2 H. dg,g 2 d,. g g 2 Therefore, the function u is continuous. Thus, by Lemmas 2. and 7. we deduce that the probability measure P N µ N fs + imh A PN,f,f 2 u f µ 2 s+imh N f, A A BM s+imh converges weakly to the distribution of the random element f 2s,ω. However, f s,ω r j0 a mjω j ωme λm+jλ s r j ωe λ s j s f 2 s, ω f s, ω r j ωe λ s j s a mωme λms r j ωe λ s j s fs, ω. Therefore, the limit measure is m H ω Ω: fs, ω A, A BM. The theorem is proved.

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