Download The Laws of Large Numbers by Pál Révész, Z. W. Birnbaum and E. Lukacs (Auth.) PDF

By Pál Révész, Z. W. Birnbaum and E. Lukacs (Auth.)

ISBN-10: 1483230554

ISBN-13: 9781483230559

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Example text

E. E(ec»), will (occasionally) be evaluated. 3. At times these two methods will be applied to truncated random variables. 2. Now we can turn to the investigation of more general results. First of all we prove a very weak necessary condition. 2) for any ε > 0. P R O O F . 2) are equivalent by the Borel-Cantelli lemma. 6. 3) otherwise. Since m(fjf) = 0 (k = 1, 2, . 3). 5) but we do not use this restriction in all cases. 6. 3. 7) ζ2*->0. P R O O F . 7) is trivial. e. 2 n _ 1 < k < 2n (n = n(k)). Then we have Ρ{|%·- ^|^2"ε} = ^1_ρ(| % η | > 2 "- 1-Ρ{|^η-^|>2"ε}^ 1 ε}-Ρ{|^|>2"- 1 ε}.

H + ■ ■ ■ + Sj Ak — { ω : Vt < x, η2 <, x, ■ ■ ■ , Vk-i <1x,Vk> x} and A = {ω : ηη > x — 2 frä} . Clearly, AkBk2)fn)^ (4 = 1 , 2 , . . , » ) . 2) together imply = P(^)^ ΣΡ(ΑΒΗ)>^-Σ 4 \k=\ 4 ) Ρ(Λ) = 4 Λ=1 k=\ \\

Xn) = = P(*lh < *i> Vi2 < *2> · · · > Vin < χη) < <π)- (h < h < Then Σ f / converges (with probability I) if and only if Σ Vi converges i=l i=l (with probability I) and ξ obeys a law of large numbers if η does the same. 3a) are interesting from the point of view of the Jaws of large numbers. 2 (KRONECKER LEMMA). / / λν λ2, . . is a monotonically increasing sequence of positive numbers tending to infinity and uv u2, . . is an arbitrary sequence of real numbers then the convergence of the series Σ —~ implies n—\ λη PROOF.

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