uniform distribution theory

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1 niform Distribution heory 4 (2009), no., 9 26 uniform distribution theory WEIGHED LIMI HEOREMS FOR GENERAL DIRICHLE SERIES. II Jonas Genys Antanas Laurinčikas ABSRAC. nder some hypotheses on the weight function and a function given by general Dirichlet series, weighted limit theorems in the sense of weak convergence of probability measures in the space of meromorphic functions are obtained. If the system of exponents of Dirichlet series is linearly independent over the field of rational numbers, then the explicit form of the limit measure is given. Communicated by Michel Weber. Introduction Let {a m : m N} be a sequence of complex numbers, and {λ m : m N} be an increasing sequence of real numbers such that lim λ m = +. hen the m series a m e λms, s = σ +it, () m= is called a general Dirichlet series. he region of convergence as well as of absolute convergence of series () is a half-plane. From H. Bohr and B. Jessen fundamental works [2], [3], the asymptotic behaviour of Dirichlet series is characterized by limit theorems in the sense of weak convergence of probability measures. heorems of such a type for general Dirichlet series were obtained in [5] [7] and [9] [3] Mathematics Subject Classification: M4, 60B0. Keywords: Dirichlet series, limit theorem, probability measure, space of analytic functions, space of meromorphic functions, topological group, weak convergence. Partially supported by Lithuanian Foundation of Studies and Science. 9

2 JONAS GENYS ANANAS LARINČIKAS his paper is a continuation of [6], where weighted limit theorems on the complex plane for general Dirichlet series were obtained. Before stating them, we recall the used hypotheses on Dirichlet series and the weight function. Suppose that the series () converges absolutely for σ > σ a and has a sum f(s). Additionally, we assume that the function f(s) can be meromorphically continued to the region σ > σ, σ < σ a, all poles in this region are included in a compact set, and that, for σ > σ, σ is not the real part of a pole of f(s), the estimates f(σ +it) = O( t a ), a = a(σ) > 0, t t 0 > 0, (2) and f(σ +it) 2 dt = O(), (3) are satisfied. Let w(t) be a positive function of bounded variation on [, ), > 0. Moreover, let = (,w) = w(t)dt. We suppose that lim (,w) = +, and that, for σ > σ, σ is not the real part of a pole of f(s), and all v R, the estimate +v +v w(t v) f(σ +it) 2 dt = O((+ v )) (4) is satisfied. Note that if w(t) = t, then the estimate (3) implies (4). Denote by B(S) the class of Borel sets of a metric space S, and define the probability measure P,σ,w (A) = w(t)i {t:f(σ+it) A} dt, A B(C), where I A denotes the indicator function of the set A, and C is the complexplane. he first theorem of [6] is the following statement. Ì ÓÖ Ñ º [6]. Suppose that σ > σ and that the function f(s) satisfies the conditions (2) and (4). hen on (C,B(C)), there exists a probability measure P σ such that the measure P,σ,w converges weakly to P σ as. 0

3 WEIGHED LIMI HEOREMS FOR GENERAL DIRICHLE SERIES. II he identification of the limit measure P σ requires some definitions and additional hypotheses on the functions f(s) and w(t). Let γ = {s C : s = } denote the unit circle on the complex plane, and Ω = γ m, m= where γ m = γ for all m N. By the ikhonov theorem, with the product topology and pointwise multiplication, the infinite dimensional torus Ω is a compact topological Abelian group. herefore, on (Ω, B(Ω)) the probability Haar measure m H can be defined, and this leads to a probability space (Ω,B(Ω),m H ). Denote by ω(m) the projection of ω Ω to the coordinate space γ m, m N. Suppose that λ m c(logm) δ (5) withsomepositive constantscandδ, andontheprobabilityspace(ω,b(ω),m H ) define the complex-valued random element f(σ, ω) by f(σ,ω) = a m ω(m)e λmσ, σ > σ. m= Note that inequality (5) is used to provein [3] that f(σ,ω) is a random element. Also, we need a condition for w(t) which generalizes the classical Birkhoff- Khintchine theorem, see, for example, [4]. Denote by EX the expectation of the random elementx. Let X(t,ω)be anarbitraryergodicprocess, E X(t,ω) <, with sample paths integrable almost surely in the Riemann sense over every finite interval. We suppose that w(t)x(t+v,ω)dt = EX(0,ω)+r (+ v ) almost surely for all v R with some α > 0, where r 0 as. If in (6) w(t) and v = 0, then (6) becomes the Birkhoff-Khintchine theorem. Let μ() = inf w(t). If t [,] w()μ () = O(), then the weight function w(t) satisfies (6) with α = (see [4]). Ì ÓÖ Ñ 2º [6]. Let σ > σ. Suppose that the system {λ m : m N} is linearly independent over the field of rational numbers, and inequality (5) holds. Moreover, suppose that the weight function w(t) satisfies (6), and, for the function f(s), estimates (2) and (4) hold. hen the probability measure P,σ,w converges weakly to the distribution P f,σ of the random element f(σ,ω) as. α (6)

4 JONAS GENYS ANANAS LARINČIKAS We recall that P f,σ is defined by P f,σ (A) = m H (ω Ω : f(σ,ω) A), A B(C). he aim of this paper is to prove weighted limit theorems in the space of meromorphic functions for the function f(s). he first limit theorem in this space for the function f(s) was obtained in [0]. Let C = C { } be the Riemann sphere, and let d(s,s 2 ) denote the spherical metric, i. e., for s,s 2,s C, d(s,s 2 ) = 2 s 2 s + s 2 + s 2 2, d(s, ) = 2, d(, ) = 0. + s 2 Denote by D the half-plane {s C : σ > σ }, and let M(D) stand for the space of meromorphic functions g : D (C,d) equipped with the topology of uniform convergence on compacta. In this topology, a sequence {g n } M(D) converges to g M(G) if sup d(g n (s),g(s)) 0 s K n for every compact subset K of D. Analytic on D functions form a subspace H(D) of M(G). Denote by meas{a} the Lebesgue measure of a measurable set A R. Ì ÓÖ Ñ 3º [0]. Suppose that the function f(s) satisfies the hypotheses (2) and (3). hen on (M(D),B(M(D))), there exists a probability measure P such that the measure meas{τ [0,] : f(s+iτ) A}, A B(M(D)), converges weakly to P as. he first attempt for identification of the limit measure P in heorem 3 was made in [3], using the hypothesis that the set {log2} {λ m : m N} is linearly independent over the field of rational numbers. In [5], the number log 2 was removed from the hypothesis. Suppose that inequality (5) is satisfied, and for s D, on the probability space (Ω,B(Ω),m H ), define the H(D)-valued random element f(s,ω) by the formula 2 f(s,ω) = a m ω(m)e λms. m=

5 WEIGHED LIMI HEOREMS FOR GENERAL DIRICHLE SERIES. II Ì ÓÖ Ñ 4º [5]. Suppose that the system {λ m : m N} is linearly independent over the field of rational numbers, and that conditions (2), (3) and (5) are satisfied. hen the measure meas{τ [0,] : f(s+iτ) A}, A B(M(D)), converges weakly to the distribution of the H(D)-valued random element f(s, ω) as. Now we state new weighted limit theorems in the space of meromorphic functions for f(s). Define P,w (A) = w(τ)i {τ:f(s+iτ) A} dτ, A B(M(D)). Ì ÓÖ Ñ 5º Suppose that conditions (2) and (4) are satisfied. hen on (M(D), B(M(D))), there exists a probabilitymeasure P w such that P,w converges weakly to P w as. he next theorem gives the explicit form for the limit measure P w in heorem 5. Ì ÓÖ Ñ 6º Suppose that the system {λ m : m N} is linearly independent over the field of rational numbers, and inequality (5) holds. Moreover, suppose that the weight function w(t) satisfies (6), and, for the function f(s), estimates (2) and (4) hold. hen the probability measure P,w converges weakly to the distribution P f of the random element f(s,ω) as. Note that the distribution P f is defined by P f (A) = m H (ω Ω : f(s,ω) A), A B(H(D)). heorems 5 and 6 show, in some sense, the regularity of the asymptotic behaviuor of the function f(s). History and bibliography on probabilistic results for zeta-functions and Dirichlet series are given in [6], the first part of our work. 2. Limit theorems for absolutely convergent Dirichlet series Since all poles of f(s) in the half-plane D are included in a compact set, their number is finite. Denote the poles of f(s) in the region D by s,...,s r. Without 3

6 JONAS GENYS ANANAS LARINČIKAS loss of generality we can assume that every of these poles has order. Define r ( ) f (s) = e λ (s j s). hen, obviously, f (s j ) = 0 for j =,...,r. herefore, the function j= f 2 (s) = f (s)f(s) is regular on D. Denote by A the number of elements of the set A. hen, for σ > σ a, the function f 2 (s) can be written in the form f 2 (s) = a m e λ j A s j ( ) A e (λ m+ A λ )s = A {,...,r} m= = r j=0 m= a m,j e (λ m+jλ )s by the absolute convergence with the coefficients a m,j = O( a m ), m N, j =,...,r. It is clear that the definition of f 2 (s), and (2) and (4), for σ > σ, imply the estimate f 2 (σ +it) = O( t a ), a = a(σ) > 0, t t 0 > 0, (7) and, for all v R, the estimate +v +v w(t v) f 2 (σ +it) 2 dt = O((+ v )). (8) Let σ 2 > σ a σ be a fixed number, v(m,n) = exp{ e (λ m λ n )σ 2 } and r g n (s) = a m,j v(m,n)e (λ m+jλ )s. j=0 m= In view of (7) and (8), in the same way as in [6], it follows that the series for g n (s) converges absolutely for σ > σ. In this section, first we consider the weak convergence of the probability measure P,n,w (A) = w(τ)i {τ:gn (s+iτ) A}dτ, A B(H(D)). For this, we recall a limit theorem on the torus Ω. 4

7 Let WEIGHED LIMI HEOREMS FOR GENERAL DIRICHLE SERIES. II Q,w (A) = w(τ)i :m N) A}dτ, A B(Ω). {τ:(e iλmτ Ä ÑÑ º [6]. On (Ω,B(Ω)), there exists a probabilitymeasure Q w such that the measure Q,w converges weakly to Q w as. If the system {λ m : m N} is linearly independent over the field of rational numbers, then the measure Q,w converges weakly to the Haar measure m H as. Ì ÓÖ Ñ 7º On (H(D),B(H(D))), there exists a probabilitymeasure P n,w such that the measure P,n,w converges weakly to P n,w as. Proof. Define the function h n : Ω H(D) by the formula r h n (ω) = a m,j v(m,n)ω(m)ω j ()e (λ m+jλ )s. j=0 m= hen the function h n is continuous, and h n ( (e iλ m τ : m N) ) = g n (s+iτ). Consequently, P,n,w = Q,w h n, and Lemma and heorem 5. of [] imply the weak convergence of P,n,w to Q w h n as. For ω Ω, let g n (s,ω) = r j=0 m= a m,j v(m,n)ω(m)ω j ()e (λ m+jλ )s. Obviously, the latter series also converges absolutely for σ > σ. Let ˆω be a fixed element of Ω, and ˆP,n,w (A) = w(τ)i {τ:gn (s+iτ,ˆω) A}dτ, A B(H(D)). Ì ÓÖ Ñ 8º Suppose that the system {λ m : m N} is linearly independent over the field of rational numbers. hen on (H(D), B(H(D))), there exists a probabilitymeasure P n such that the measures P,n,w and ˆP,n,w converge weakly to P n as. Proof. Since the system {λ m : m N} is linearly independent over the field of rational numbers, by the second part of Lemma and the proof of heorem 7 we have that P,n,w converges weakly to m H h n as. 5

8 JONAS GENYS ANANAS LARINČIKAS Define ĥn : Ω H(D) by the formula r ĥ n (ω) = a m,j v(m,n)ˆω(m)ˆω j ()ω(m)ω j ()e (λ m+jλ )s. j=0 m= hen, in the same way as in the case of P,n,w, we obtain that ˆP,n,w converges weakly to m H ĥ n as. Now we take h : Ω Ω defined by h(ω) = ωˆω. hen, clearly, ĥn(ω) = h n (h(ω)). Since the Haar measure is invariant, the equality m H ĥ n = m H (h n (h)) = (m H h )h n = mh n holds, and the theorem is proved. 3. Approximation by the mean For s D and ω Ω, define r f 2 (s,ω) = a m,j ω(m)ω j ()e (λ m+jλ )s. j=0 m= In this section, we approximate f 2 (s) by g n (s) as well as f 2 (s,ω) by g n (s,ω) in the mean. his is necessary to deduce heorems 5 and 6 from heorems 7 and 8, respectively. Ì ÓÖ Ñ 9º Let K be a compact subset of the half-plane D. hen lim limsup w(τ)sup f 2 (s+iτ) g n (s+iτ) dτ = 0. n s K Proof. Let σ 2 be the same as in definition of g n (s), and l n (s) = s ( ) s Γ e λns, n N. σ 2 hen the function g n (s) can be written in the form in [5] 6 g n (s) = 2πi σ 2 σ 2 +i σ 2 i f 2 (s+z)l n (z) dz z.

9 WEIGHED LIMI HEOREMS FOR GENERAL DIRICHLE SERIES. II From this, for σ 3 > σ and σ 3 < σ, using the residue theorem we derive g n (s) = 2πi σ 3 σ+i σ 3 σ i f 2 (s+z)l n (z) dz z +f 2(s). (9) Let L be a simple closed contour lying in D and enclosing the set K. Denote by δ the distance of L from the set K. hen by the Cauchy integral formula sup f 2 (s+iτ) g n (s+iτ) f 2 (z +iτ) g n (z +iτ) dz. s K 2πδ Hence, we obtain δ L dz L δ w(τ)sup f 2 (s+iτ) g n (s+iτ) dτ s K +Imz +Imz sup σ+iu L +u L w(τ Imz) f 2 (Rez +iτ) g n (Rez +iτ) dτ +u w(t u) f 2 (σ +it) g n (σ +it) dt, (0) where L denotes the length of the contour L. Suppose that min{σ : s K} = σ +ε, ε > 0, and max{σ : s K} = A. We put σ 3 = σ + ε 2. he contour L can be chosen so that, for s L, the inequalities σ σ + 3ε 4 and δ ε 4 hold. In view of (9), f 2 (σ +it) g n (σ +it) f 2 (σ 3 +it+iτ) l n (σ 3 σ +iτ) dτ. herefore, the Cauchy-Schwarz inequality and (8) yield +u +u l n (σ 3 σ +iτ) w(t u) f 2 (σ +it) g n (σ +it) dt +u+τ +u+τ w(t u τ) f 2 (σ 3 +it) dtdτ 7

10 l n (σ 3 σ+iτ) JONAS GENYS ANANAS LARINČIKAS +u+τ +u+τ l n (σ 3 σ +iτ) (+ u + τ ) 2 dτ w(t u τ) f 2 (σ 3 +it) 2 dt 2 dτ l n (σ 3 σ +iτ) (+ τ )dτ, because u is bounded by a constant. his together with (0) show that w(τ)sup f 2 (s+iτ) g n (s+iτ) dτ s K as n. sup σ ε 4 l n (σ +it) (+ t )dt = o() Ì ÓÖ Ñ 0º Suppose that the system {λ m : m N} is linearly independent over the field of rational numbers. Let K be a compact subset of the half-plane D. hen, for almost all ω Ω, lim limsup n w(τ)sup s K f 2 (s+iτ,ω) g n (s+iτ,ω) dτ = 0. Proof. Let a τ = {e iτλ m : m N}, τ R, and define φ τ (ω) = a τ ω for ω Ω. hen {φ τ : τ R} is a one-parameter group of measurable measure preserving transformations on Ω. In [6], Lemma 5, it was proved that the one-parameter group {φ τ : τ R} is ergodic. From this and (6), similarly to the proof of Lemma 2 of [6] we obtain that, for σ > σ, +v +v w(t v) f 2 (σ +it,ω) 2 dt (+ v ) α for almost all ω Ω and all v R. sing this estimate instead of (8) and repeating the arguments of the proof of heorem 9, we obtain the assertion of the theorem. 8

11 WEIGHED LIMI HEOREMS FOR GENERAL DIRICHLE SERIES. II 4. Limit theorems for the function f 2 (s) his section is devoted to limit theorems in the space of analytic functions for the function f 2 (s). On (H(D)), B(H(D)), define two probability measures and, for ω Ω, P,f2,w(A) = ˆP,f2,w(A) = w(τ)i {τ:f2 (s+iτ) A}dτ w(τ)i {τ:f2 (s+iτ,ω) A}dτ. Ì ÓÖ Ñ º Suppose that the hypotheses of heorem 5 are satisfied. hen on (H(D),B(H(D))), there exists a probability measure P f2,w such that the measure P,f2,w converges weakly to P f2,w as. Proof. By heorem 7, we have that P,n,w converges weakly to some probability measure P n,w on (H(D),B(H(D))) as. Let X n,w (s) be a H(D)- D valued random element having the distribution P n,w, and denote the convergence in distribution. Moreover, let θ be a random variable defined on a certain probability space (ˆΩ,B(ˆΩ),P) such that Define hen we have from heorem 7 that P(θ A) = w(τ)i A dτ, A B(R). X,n,w (s) = g n (s+iθ ). X,n,w (s) D X n,w(s). () In the next step we prove that the family of probability measures {P n,w : n N} is tight. It is well known, see, for example, Lemma.7. of [7], that there exists a sequence {K l : l N} of compact subsets of D such that D = K l, K l K l+, and if K is a compact subset of D, then K K l for some l. hen ρ(g,g 2 ) = l= 2 l sup s K l g (s) g 2 (s) + sup s K l g (s) g 2 (s), g,g 2 H(D), l= 9

12 JONAS GENYS ANANAS LARINČIKAS is a metric on H(D) which induces its topology of uniform convergence on compacta. For every M l > 0, l N, we have w(τ)i {τ: sup g n (s+iτ) >M l }dτ s K l w(τ) sup g n (s+iτ) dτ. (2) M l s K l Moreover, by heorem 9 and Cauchy integral formula sup lim sup w(τ) sup g n (s+iτ) dτ n N s K l sup lim sup n N +sup lim sup n N +limsup w(τ) sup s K l f 2 (s+iτ) g n (s+iτ) dτ+ w(τ) sup f 2 (s+iτ) dτ s K l w(t) f 2 (σ +it) 2 dt with some σ > σ. herefore, this and (8) show that sup lim sup w(τ) sup g n (s+iτ) dτ R l <, (3) n N s K l l N. Now we take M l = M l,ε = R l 2 l ε, where ε > 0 is arbitrary number. hen, in view of (2) and (3), ( ) lim supp sup X,n,w (s) > M l = s K l 20 = w(τ)i {τ: sup g n (s+iτ) >M l }dτ ε s K l 2l, l N, n N. (4) 2

13 WEIGHED LIMI HEOREMS FOR GENERAL DIRICHLE SERIES. II Relation () implies D sup X,n,w (s) sup X n,w (s), l N, n N. s K l s K l herefore, by (4), ( ) P sup X n,w (s) > M l s K l ε 2l, l N, n N. (5) Define H ε = {g H(D) : sup g(s) M l,ε, l N}. s K l Since the set H ε is uniformly bounded on every compact of D, it is a compact subset of H(D). Moreover, in view of (5), P(X n,w (s) H ε ) ε = ε, n N. 2l his shows that the family {P n,w : n N} is tight, therefore, by the Prokhorov theorem, see, for example, [], heorem 6., it is relatively compact. Hence, there exists a sequence {P nk,w} {P n,w } such that P nk,w converges weakly to a certain probability measure P f2,w on (H(D),B(H(D))) as k. In other words, D X nk,w(s) P f 2,w. (6) k Define once one H(D)-valued random element X,w (s) by hen, by heorem 9, for every ε > 0, l= X,w (s) = f 2 (s+iθ ). lim limsup P(ρ(X,w (s),x,n,w (s)) ε) n lim limsup w(τ)ρ(f 2 (s+iτ),g n (s+iτ))dτ = 0. n ε his, (), (6) and heorem 4.2 of [] show that and the theorem is proved. X,w (s) D P f 2,w, (7) Ì ÓÖ Ñ 2º Suppose that the hypotheses of heorem 6 are valid. hen the probability measures P,f2,w and ˆP,f2,w both converge weakly to the same probability measure on (H(D),B(H(D))) as. 2

14 JONAS GENYS ANANAS LARINČIKAS Proof. In view of heorem, it remains to show that the measure ˆP,f2,w also converges weakly to the measure P f2,w as. We also preserve the notation of the proof of heorem. By heorem 8, the measures P,n,w and ˆP,n,w converge weakly to some probability measure P n,w on (H(D),B(H(D))) as. Let ˆX,n,w (s) = g n (s+iθ,ω). hen we have that D ˆX,n,w (s) X n,w(s), (8) where the H(D)-valued random element X n,w (s) was defined in the proof of heorem. he relation (7) shows that the measure P f2,w is independent of the choice of the sequence {P nk,w}. his and the relative compactness of the family {P n,w } imply the relation D X n,w (s) P f 2,w. (9) n Let ˆX,w (s) = f 2 (s+iθ,ω). hen, by heorem 0, for every ε > 0 and almost all ω Ω, ( lim limsup P ρ( ˆX,w (s), ˆX ),n,w (s)) ε n lim limsup w(τ)ρ(f 2 (s+iτ,ω),g n (s+iτ,ω))dτ = 0. n ε his, (8) and (9) together with heorem 4.2 of [] again imply the weak convergence of ˆP,f2,w to P f2,w as. Denote by P f2 the distribution of the random element f 2 (s,ω). Ì ÓÖ Ñ 3º Suppose that the hypotheses of heorem 6 are valid. hen the probability measure P,f2,w converges weakly to P f2 as. Proof. In view of heorem 2, it suffices to show that the measure P f2,w coincides with P f2. Let A B(H(D)) be an arbitrary fixed continuity set of the limit measure P f2,w in heorem 2. hen heorem 2 and an equivalent of weak convergence of probability measures, see heorem 2. of [], imply 22 lim w(τ)i {τ:f2 (s+iτ,ω) A}dτ = P f2,w(a). (20)

15 WEIGHED LIMI HEOREMS FOR GENERAL DIRICHLE SERIES. II On the probability space (Ω,B(Ω),m H ), define the random variable θ by { if f2 (s,ω) A, θ(ω) = 0 if f 2 (s,ω) A. hen, obviously, Eθ = θ(ω)dm H = m H (ω Ω : f 2 (s,ω) A) = P f2 (A). (2) Ω Since the one-parameter group {φ τ : τ R} is ergodic, the random process θ(φ τ (ω)) is ergodic, too. Hence, using the hypothesis (6) with v = 0, we have that, for almost all ω Ω, lim From the definitions of θ and φ τ (ω) it follows that w(τ)θ(φ τ (ω))dτ = w(τ)θ(φ τ (ω))dτ = Eθ. (22) w(τ)i {τ:f2 (s,φ τ (ω)) A}dτ = = w(τ)i {τ:f2 (s+iτ,ω) A}dτ. herefore, taking into account (2) and (22), we obtain that, for almost all ω Ω, lim w(τ)i {τ:f2 (s+iτ,ω) A}dτ = P f2 (A). hus, by (20), for all continuity sets A of P f2,w, P f2,w(a) = P f2 (A). Hence, P f2,w(a) = P f2 (A) for all A B(H(D)). he theorem is proved. First we observe that 5. Proof of theorems 5 and 6 f (s) = r j= ( e λ (s j s) ) = r b m e λ ms m=0 23

16 JONAS GENYS ANANAS LARINČIKAS is a Dirichlet polynomial with some coefficients b m and exponents mλ. herefore, an application of Lemma shows that the probability measure w(τ)i {τ:f (s+iτ) A}dτ, A B(H(D)), converges weakly to the distribution P f the H(D)-valued random element f (s,ω) = r j= ( ω()e λ (s j s) ) = as. Now let H 2 (D) = H(D) H(D), and r b m ω m ()e λ ms m=0 P,f,f 2,w(A) = w(τ)i {τ:(f (s+iτ),f 2 (s+iτ)) A}dτ, A B(H 2 (D)), Ä ÑÑ 2º Suppose that the hypotheses of heorem 5 are satisfied. hen on (H 2 (D),B(H 2 (D))) thereexists aprobabilitymeasurep f,f 2,wsuchthat P,f,f 2,w converges weakly to P f,f 2 as. Proof. Let, for g j = (g j,g j2 ) H 2 (D), j =,2, ρ(g,g 2 ) = max j 2 (ρ(g,g 2 ),ρ(g 2,g 22 )). hen ρ is a metric on H 2 (D) inducing its topology. sing this metric and repeating the arguments of the proof of heorem with obvious changes, we obtain the statement of the lemma. Define the H 2 (D)-valued random element F(s,ω) by and denote by P F its distribution. F(s,ω) = (f (s,ω),f 2 (s,ω)), Ä ÑÑ 3º Suppose that the hypotheses of heorem 6 are valid. hen the measure P,f,f 2,w converges weakly to P F as. Proof. he lemma is obtained in the same way as heorem 3 with using the metric ρ. 24

17 WEIGHED LIMI HEOREMS FOR GENERAL DIRICHLE SERIES. II Proof of heorem 5. Define the function h : H 2 (D) M(D), by the formula h(g,g 2 ) = g 2, g,g 2 H(D). g Since the metric d satisfies ( d(g,g 2 ) = d, ), g g 2 the function h is continuous. herefore, by Lemma 2 and heorem 5. of [], the measure P,w = P,f,f 2,wh converges weakly to the measure P f,f 2,wh as. Proof of heorem 6. Similarly to the proof of heorem 5, using Lemma 3, we find that the measure P f,f 2,w converges weakly to the measure P F h as. However, ( P F h (A) = m H ω Ω : f ) 2(s,ω) f (s,ω) A,A B(M(D)). Since f 2 (s,ω) = = r j=0 m= a m,j ω j ()ω(m)e (λ m+jλ )s = r ( ω()e ) λ (s j s) a m ω(m)e λms, j= hence we have that P,w converges weakly to the distribution of the random element f(s,ω) as. m= REFERENCES [] BILLINGSLEY, P.: Convergence of Probability Measures, John Wiley & Sons Inc, New York, 968. [2] BOHR, H. JESSEN, B.: Über die Wertverteilung der Riemannschen Zetafunktion, Erste Mitteilung, Acta Math. 54 (930), 35. [3] BOHR, H. JESSEN, B.: Über die Wertverteilung der Riemannschen Zetafunktion, Zweite Mitteilung, 58 (932), 55. [4] CRAMÉR, H. LEADBEER, M. R.: Stationary and Related Stochastic Processes, John Wiley, New York, 967. [5] GENYS, J. LARINČIKAS, A.: Value distribution of general Dirichlet series. IV, Lith. Math. J. 43(3) (2003),

18 JONAS GENYS ANANAS LARINČIKAS [6] GENYS, J. LARINČIKAS, A.: Weighted limit theorems for general Dirichlet series, niform Distribution heory 2 (2007), no. 2, [7] GENYS, J. LARINČIKAS, A. ŠIAČIŪNAS, D.: Value distribution of general Dirichlet series. VII, Lith. Math. J. 46(2) (2006), [8] LARINČIKAS, A.: Limit heorems for the Riemann Zeta-Function, Kluwer Academic Publishers, Dordrecht - Boston - London, 996. [9] LARINČIKAS, A.: Value-distribution of general Dirichlet series, in: Prob. heory and Math. Stat., Proc. 7th Vilnius Conference,998 (B. Grigelionis et al. Eds.), VSP/trecht, EV, Vilnius, 999, pp [0] LARINČIKAS, A.: Value distribution of general Dirichlet series. II, Lith. Math. J. 4(4) (200), [] LARINČIKAS, A.: Limit theorems for general Dirichlet series, heory of Stochastic Processes 8(24) (2002), no. 3-4, [2] LARINČIKAS, A.: Value distribution of general Dirichlet series. VIII, Chebyshevskii Sbornik 7 (2006), no. 2, [3] LARINČIKAS, A. SCHWARZ, W. SEDING, J.: Value distribution of general Dirichlet series. III, in: Anal. Probab. Methods Numb. heor., Proc. hird International Conference in Honor of J. Kubilius, Palanga, 200 (A. Dubickas et al.eds.), EV, Vilnius, 2002, pp [4] LARINČIKAS, A. MISEVIČIS, G.: Weighted limittheorem for the Riemann zeta-function in the space of analytic functions, Acta Arith. LXXVI (996), no. 4, Received September 8, 2008 Accepted January 2, 2009 Jonas Genys Faculty of Mathematics and Informatics Šiauliai niversity P. Višinskio 9 L 7756 Šiauliai LIHANIA genys@fm.su.lt Antanas Laurinčikas Faculty of Mathematics and Informatics Vilnius niversity Naugaurduko 24 L Vilnius LIHANIA antanas.laurincikas@maf.vu.lt 26