<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>X</italic>1,<italic>X</italic>2,... be i.i.d. absolutely continuous random variables, let <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0015_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mi>k</mi></msub><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mrow><msub><mrow><mi>X</mi></mrow><mi>j</mi></msub></mrow></mrow></math><tex-math>{S_k} = \sum\nolimits_{j = 1}^k {{X_j}}</tex-math></alternatives></inline-formula> (mod 1) and let <italic>D*</italic><italic>N</italic> denote the star discrepancy of the sequence (<italic>S</italic><italic>k</italic>)1≤<italic>k</italic>≤<italic>N</italic>. We determine the limit distribution of <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0015_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mi>N</mi></msqrt><msubsup><mrow><mi>D</mi></mrow><mi>N</mi><mo>*</mo></msubsup></mrow></math><tex-math>\sqrt N D_N^*</tex-math></alternatives></inline-formula> and the weak limit of the sequence <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0015_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mi>N</mi></msqrt><mrow><mo>(</mo><mrow><msub><mrow><mi>F</mi></mrow><mi>N</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>-</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>\sqrt N \left( {{F_N}(t) - t} \right)</tex-math></alternatives></inline-formula> in the Skorohod space <italic>D</italic>[0, 1], where <italic>F</italic><italic>N</italic> (<italic>t</italic>) denotes the empirical distribution function of the sequence (<italic>S</italic><italic>k</italic>)1≤<italic>k</italic>≤<italic>N</italic>.</abstract>

Let X1,X2,... be i.i.d. absolutely continuous random variables, let Sk=∑j=1kXj{S_k} = \sum\nolimits_{j = 1}^k {{X_j}} (mod 1) and let D*N denote the star discrepancy of the sequence (Sk)1≤k≤N. We determine the limit distribution of NDN*\sqrt N D_N^* and the weak limit of the sequence N(FN(t)-t)\sqrt N \left( {{F_N}(t) - t} \right) in the Skorohod space D[0, 1], where FN (t) denotes the empirical distribution function of the sequence (Sk)1≤k≤N.

On the Discrepancy of Random Walks on the Circle

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"On the Discrepancy of Random Walks on the Circle"}

Let X1,X2,... be i.i.d. absolutely continuous random variables, let Sk=∑j=1kXj{S_k} = \sum\nolimits_{j = 1}^k {{X_j}} (mod 1) and let D*N denote the star...

On the Discrepancy of Random Walks on the Circle

Data publikacji: 27 mar 2020

Zakres stron: 73 - 86

Otrzymano: 24 sie 2018

Przyjęty: 12 mar 2019

DOI: https://doi.org/10.2478/udt-2019-0015

Słowa kluczowe
i.i.d. sums mod 1, empirical distribution, discrepancy, weak convergence

© 2019 Alina Bazarova et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On the Discrepancy of Random Walks on the Circle

Data publikacji: 27 mar 2020

Zakres stron: 73 - 86

Otrzymano: 24 sie 2018

Przyjęty: 12 mar 2019

DOI: https://doi.org/10.2478/udt-2019-0015

Słowa kluczowei.i.d. sums mod 1, empirical distribution, discrepancy, weak convergence

© 2019 Alina Bazarova et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Słowa kluczowe
i.i.d. sums mod 1, empirical distribution, discrepancy, weak convergence