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a limit theorem for the hurwitz zeta-function in the space of analytic ...

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R. Macaitienė 87<br />

Aply<strong>in</strong>g <strong>the</strong> Chebyshev <strong>in</strong>equality once more and Lemma 7, we deduce that <strong>for</strong><br />

every ε > 0<br />

lim lim<br />

(<br />

ρ(ζ(s + iτ, α), ζ2,n (s + iτ, α)) ε )<br />

Let<br />

n→∞ T →∞ ντ T<br />

∫ T<br />

lim<br />

lim<br />

n→∞ T →∞<br />

1<br />

εT<br />

0<br />

ρ(ζ(s + iτ, α), ζ 2,n (s + iτ, α))dτ = 0.<br />

Y T (s, α) = ζ(s + iT θ, α).<br />

Then <strong>the</strong> relation (17) can be written <strong>in</strong> <strong>the</strong> <strong>for</strong>m<br />

lim<br />

n→∞<br />

(17)<br />

lim P(ρ(X T,n(s, α), Y T (s, α)) ε) = 0. (18)<br />

T →∞<br />

Let {P 1 n ′} bet a subsequence <strong>of</strong> {P 1 n} which converges weakly to some measure<br />

P 1 . S<strong>in</strong>ce <strong>the</strong> <strong>space</strong> H(D) is separable, by (15) we have that<br />

X n ′<br />

D<br />

−→ P 1 , n ′ → ∞.<br />

From this and <strong>the</strong> relations (15) and (18), we deduce that<br />

Y T<br />

D<br />

−→ P 1 , T → ∞. (19)<br />

This relation is equivalent to <strong>the</strong> weak convergence <strong>of</strong> P T to P 1 . From (19) <strong>in</strong> view<br />

<strong>of</strong> <strong>the</strong> relative compactness <strong>of</strong> <strong>the</strong> family {Pn} 1 and <strong>of</strong> <strong>the</strong> assertion (The relation<br />

P n ⇒ P is true if and only if every subsequence P n ′ conta<strong>in</strong>s ano<strong>the</strong>r subsequence<br />

P n ′′ such that P n ′′ ⇒ P ) we get<br />

X n<br />

D<br />

−→ P 1 , n → ∞. (20)<br />

and<br />

Repeat<strong>in</strong>g <strong>the</strong> analogous reason<strong>in</strong>g <strong>for</strong> <strong>the</strong> random elements<br />

˜X T,n (s, ω 1 , α) = ζ 2,n (s + iT θ, ω 1 , α)<br />

Ỹ T (s, ω 1 , α) = ζ(s + iT θ, ω 1 , α)<br />

and apply<strong>in</strong>g Lemma 9 and (20), we obta<strong>in</strong> that <strong>the</strong> measure Q T converges weakly<br />

to P 1 as T → ∞. The lemma is proved.<br />

Pro<strong>of</strong> <strong>of</strong> Theorem 2. Lemma 10 asserts that <strong>the</strong> measures P T and Q T converge<br />

weakly to some measure P 1 as T → ∞ simultaneously. It rema<strong>in</strong>s to prove that<br />

P 1 = P ζ .<br />

Let A ∈ B(H(D)) be a cont<strong>in</strong>uity set <strong>of</strong> P 1 . Then by Lemma 10 we have that<br />

lim<br />

T →∞ ντ T (ζ(s + iτ, ω 1 , α) ∈ A) = P 1 (A). (21)

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