C-19 (B) Development of stochastic numerics and probability theory via lacunary series (FUKUYAMA KATUSI)

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1 C (B) Development of stochastic numerics and probability theory via lacunary series (FUKUYAMA KATUSI) discrepancy Hardy-Littewood-Pólya 3,200, ,200,000 2,600, ,600,000 2,400, ,000 3,20,000 2,500, ,000 3,250,000 0,700,000,470,000 2,70,000 discrepancyriesz-raikov. Weyl n k+ n k > c > 0 { n k x } discrepancy sup 0 a <a< N N [a,a)( n k x ) (a a ) ; k discrepancy D N ({n k x}) 0 Khinchin Kesten {k} Philipp Hadamard n k+ /n k > q > {n k } 4 < ND N ({n k x}) C q 2N log log N a.e.,

2 Erdős-Gál p q Pólya Pólya < q < < q τ {q i... qi τ τ i,..., i τ 0,, 2,... } Philipp (994) Hardy-Littlewood-Pólya Hadamard 2. discrepancy R. C. Baker 979 Hardy-Littlewood- Hardy-Littlewood- discrepancy 3. discrepancy discrepancy splitting 4. () discrepancy θ > sup σ θ,0,a,. 0 a< θ r / Q (r N), 2, θ r p/q (p, q N, r min{k N θ k Q}, gcd(p, q) ) 2 pq + 2N log log N 2 pq, p q 2 pq + pq, 2 p + p, p 4 q p 2, q (p + )!/(p 3)! 2(p ) 2, 42 9, (2) Hardy-Littlewood-Pólya discrepancy Hardy-Littlewood-Pólya C q,...,q τ ND N ({n k x}) C q,...,q 2N log log N τ,. {n k } C q,...,q τ 2 ( τ i ) /2 q i + q i q ( τ ) /2 q i + C q,...,qτ 2 q i 2 i2 (3) ( τ i ) /2 q i +. q i {n k } N σ ND N {n σ(k) x} 2N log log N 2 a.e. {n k x} discrepancy (4) n k+ n k 5 ND N {n k x} σ(x) σ(x) σ 2 (x) (4x [0,/4) (x) + [/4,3/4) (x) + 4( x) [3/4,) (x))/9 + /24.

3 5. [ ] ( 25 () V. Belitsky, Y. Higuchi, N. Konno, N. Sugimine, Analyticity of survival probability for contact process around t 0, Analysis () K. Fukuyama, The law of the iterated logarithm for the discrepancies of a permuta- and its Applications 5 (2007) tion of {n k x}, Acta Mathematica Hungarica, to appear (2) H. Sugita &S. Takanobu, The probability of two F q -polynomials to be coprime, Probability and number theory Kanazawa 2005, (2) K. Fukuyama & K. Nakata, A metric discrepancy result for the Hardy-Littlewood- Adv. Stud. Pure Math., 49 (2007) 455 Pólya sequences, Monatshefte für Mathematik, to appear 478 (3) S. Ogawa, Noncausal stochastic calculus revisited (3) K. Fukuyama, A law of the iterated logarithm for discrepancies: non-constant - around the so-called Ogawa integral. Advances in deterministic and stochastic analysis, , World Sci. Publ., Hackensup Monatshefte für Mathematik, to appear sack, NJ, (2007). (4) K. Fukuyama & Y. Komatsu, A law of large (4) S. Ogawa, K. Wakayama, On a real-time numbers for arithmetic functions, Proceedings of the American Mathematical Soci- scheme for the estimation of volatility. Monte Carlo Methods Appl. 3 (2007) no. 2, 99 ety, 37 (2009) (5) K. Fukuyama & S. Takahashi, On it (5) S. Ogawa, M. Pontier Pricing rules under distribution of trigonometric sums, Revue asymmetric information, ESAIM Probab. Roumaine des Mathematiques Pures et Appliques, 53 (2008) 9-24 Stat. (2007) (6) K. Fukuyama & Y. Ueno, On the central it theorem and the law of the iterated logarithm, Statistics & Probability Letters, 78 (2008) (7) K. Fukuyama & Y. Takeuchi, The law of the iterated logarithm for subsequences: a (6) K. Fukuyama, Gap series and functions of bounded variation, Acta Mathematica Hungarica, 0 (2006) (7) H. Sugita, Numerical integration for complicated functions and random sampling, Sugaku Expositions 9 (2006) simple proof, Lobachevskii Jour. Math., 29 (2008) (8) Y. Hamana, On the range of pinned random (8) K. Fukuyama, The law of the iterated logarithm for discrepancies of {θ n x}, Acta Mathematica Hungarica, , (9) K. Fukuyama & R. Kondo, On recurrence property of Riesz-Raikov sums, Lobachevskii Jour. Math., 26 (2007) (0) K. Fukuyama, On lacunary trigonometric product, Probability and Number Theory Kanazawa 2005, Advanced Studies in walks, Tohoku Math. J. (2) 58 (2006) no. 3, (9) Y. Hamana, A remark on the range of three dimensional pinned random walks, Kumamoto J. Math. 9(2006) (20) S. Ogawa, K. Wakayama, On a real-time scheme for the estimation of volatility, Mem. Inst. Sci. Engrg. Ritsumeikan Univ. 65 (2006) Pure Mathematics, 49 (2007)

4 (2) J. Akahori, K. Yasutomi, T. Yokota, Backwardation in Asian option prices, International Journal of Innovative Computing, Informations and Control, (2005) (22) Y. Hishida K. Yasutomi, On asymptotic behavior of the prices of Asian options, Asia-Pacific Financial Markets, 2 (2005) (23) Y. Hamana, Limit theorems for the Wiener sausage, Sugaku Expositions 8 (2005) no., (24) S. Ogawa, Introduction to the discrete approximation of stochastic differential equations, Sugaku Expositions 8 (2005) no., (25) S. Kanagawa, S. Ogawa Numerical solutions of stochastic differential equations and their applications, Sugaku Expositions 8 (2005) no., [ ] ( 7 () Hardy-Littlewood- Pólya discrepancy, (2) K. Fukuyama: On the law of the iterated logarithm for {n k x}: non-constant sup, Random matrices, special functions and related topics Nov , (3) : {n k x} discrepancy, (4),, (5) K. Fukuyama: On the law of the iterated logarithm for discrepacies of {θ n x}, Workshop on number theory and probability, Oct , (6) : {n k x} discrepancy,, (7) K. Fukuyama: On the law of the iterated logarithm for discrepacies of {θ n x}, Rencontres Mathématiques en Rouen 2007, Université de Rouen, France, June , (8) {θ n x} Discrepancy,, (9) V. Belitsky,, Analyticity of survival probablity for contact process around t 0,, (0) {θ n x} Discrepancy, () : II,, (2) : digamma, (3) K. Fukuyama: On asymptotic properties of gap series, 8 June 2006, Workshop On Probability With Applications, National Taiwan University, Taipei, Taiwan, June ,

5 (4) (5) :,, 2005, , (6) K. Fukuyama: On the law of the iterated logarithm for gap series, Department of Mathematics, Hokkaido Univ., Harmonic Analysis and its Applications at Sapporo, Organizers: A.Miyachi (Tokyo Woman s Christian Univ.), K.Tachizawa (Hokkaido Univ.) August 22, 2005, 0:45 :45, (7) K. Fukuyama: On the law of the iterated logarithm for gap series. International Conference on Probability and Number Theory 2005 (P&NT05,June 20 - June 24, 2005), Kanazawa, Japan, June 23, 2005 [] ( 3 (2) (HIGUCHI YASUNARI) : (YAMAZAKI TADASHI) : (3) (SUGITA HIROSHI) : (OGAWA SHIGEYOSHI) : (HAMANA YUJI) : (YASUTOMI KENJI) : () Probability and number theory Kanazawa Proceedings of the International Conference held in Kanazawa, June 20 24, Edited by Shigeki Akiyama, Kohji Matsumoto, Leo Murata and Hiroshi Sugita. Advanced Studies in Pure Mathematics, 49. Mathematical Society of Japan, Tokyo, xviii+558 pp. ISBN: (2),,, 2006, 239pp. (3),,, 2006, 224pp. 6. () (FUKUYAMA KATUSI) :

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