ABSTRACT
Let https://www.w3.org/1998/Math/MathML"> g ( δ ) = 2 δ . ln ( 1 / δ ) , 0 < δ < 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_217_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Let https://www.w3.org/1998/Math/MathML"> 0 < ∞ < 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_217_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Define https://www.w3.org/1998/Math/MathML"> E n = { max 1 ≤ j ≤ 2 n | B ( j / 2 n ) − B ( ( j − 1 / 2 n ) | / g ( 1 / 2 n ) ≤ 1 − θ } , n = 1 , 2 , ... https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_217_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> We shall prove that https://www.w3.org/1998/Math/MathML"> ∑ n P ( E n ) < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_217_4_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and then it will follow from the Borel-Cantelli Lemma that https://www.w3.org/1998/Math/MathML"> P ( E n , i . o ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_217_5_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> or equivalently that for a.e.ω, there exists a finite positive integer N(ω) such that for all n > N(ω), we have https://www.w3.org/1998/Math/MathML"> max 1 ≤ j ≤ 2 n | B ( j / 2 n , ω ) − B ( ( j − 1 ) / 2 n , ω | / g ( 1 / 2 n ) > 1 − θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_217_6_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> which would imply that for such ω, https://www.w3.org/1998/Math/MathML"> lim sup h ↓ 0 max 0 ≤ t ≤ 1 − h | B ( t + h , ω ) − B ( t , ω ) | / g ( h ) ≥ 1 − θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and hence letting θ ↓ 0. through rationals (Note that P(Fn ) = 1, n = 1,2, ... implies P(⋂ n Fn ) = 1), we would get the first half of the Levy modulus of continuity theorem https://www.w3.org/1998/Math/MathML"> lim sup h ↓ 0 max 0 ≤ t ≤ 1 − h | B ( t + h ) − B ( t ) | / g ( h ) ≥ 1 a . s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> To prove the summability of P(En ), we note that https://www.w3.org/1998/Math/MathML"> P ( E n ) = ( 1 − ξ n ) 2 n ≤ exp ( − 2 n ξ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in view of the independence of the events En , where https://www.w3.org/1998/Math/MathML"> ξ n = P ( | B ( 1 / 2 n ) | > g ( 1 / 2 n ) 1 − θ = 1 − Φ ( x n ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_4_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where Φ(x) is the standard normal distribution function and https://www.w3.org/1998/Math/MathML"> x n = g ( 1 / 2 n ) 1 − θ .2 n / 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_5_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Now, for any x > 0, we have using integration by parts, https://www.w3.org/1998/Math/MathML"> 1 − Φ ( x ) = ( 2 π ) − 1 / 2 ∫ x ∞ exp ( − u 2 / 2 ) d u = ( 2 π ) − 1 / 2 ( ∫ x ∞ ( 1 / u ) . u . exp ( − u 2 / 2 ) d u = ( 2 π ) − 1 / 2 ( exp ( − x 2 / 2 ) / x − ∫ x ∞ ( 1 / u 2 ) exp ( − u 2 / 2 ) d u ) ≥ ( 2 π ) − 1 / 2 ( exp ( − x 2 / 2 ) / x − x − 2 ∫ x ∞ exp ( − u 2 / 2 ) d u ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_6_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and this inequality can also be expressed as https://www.w3.org/1998/Math/MathML"> ( 1 + 1 / x 2 ) ( 1 − Φ ( x ) ) ≥ ( 2 π ) − 1 / 2 x − 1 . exp ( − x 2 / 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_7_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> or equivalently, https://www.w3.org/1998/Math/MathML"> 1 − Φ ( x ) ≥ x 1 + x 2 . exp ( − x 2 / 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_8_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Thus, https://www.w3.org/1998/Math/MathML"> ξ n ≥ x 1 + x n 2 exp ( − x n 2 / 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_9_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> x n 2 / 2 = g ( 1 / 2 n ) 2 ( 1 − θ ) .2 n − 1 = 2. ( 1 / 2 n ) . log ( 2 n ) .2 n − 1 ( 1 − θ ) = n ( 1 − θ ) log ( 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_218_10_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and hence, https://www.w3.org/1998/Math/MathML"> x n → ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_219_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and further, https://www.w3.org/1998/Math/MathML"> exp ( − x n 2 / 2 ) = 2 − n ( 1 − θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_219_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> so that https://www.w3.org/1998/Math/MathML"> exp ( − 2 n ξ n ) ≥ Κ n . exp ( − 2 n θ ) , K n = ( 2 π ) − 1 / 2 x n 1 + x n 2 ≤ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_219_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for large n and hence https://www.w3.org/1998/Math/MathML"> ∑ n exp ( − 2 n ξ n ) ≤ ∑ n exp ( − 2 n θ ) < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353430/7c6b2267-357d-43f3-8407-94415ee056ee/content/math7_219_4_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> from the desired summability of P(En ) follows.
