# p-variation calculation and application

### Description

This package deals with p-variation for the sample (i.e. the sequence of data values). It gives opportunity to calculate the p-variation for the sample – this is the main purpose of this package. Nonetheless, it could be used to calculate p-variation for arbitrary piecewise monotonic function as well. Moreover, the package includes one example of practical application of the p-variation.

### Details

Package: | pvar |

Type: | Package |

Version: | 2.2 |

Date: | 2016-05-17 |

License: | GPL-2 |

Institution: | Vilnius University Faculty of Mathematics and Informatics |

This package is about p-variation. It deals with p-variation of a finite sample data values. To be precise, lets star with the definitions. Originally p-variation is defined for a functions.

For a function *f:[0,1] -> R* and *0 < p < ∞*
p-variation is defined as

*
v_p(f) = sup { ∑ |f(t_i) - f(t_{i-1})|^p : 0=t_0<t_1<…<t_m=1, m>=1}
*

Analogically, for a sequences of values *X_0, X_1,..., X_n*, the p-variation is defined as

*
v_p({X_i}_{i=0}^n) = max { ∑ |X_{j_i}-X_{j_{i-1}}|^p :0=j_0<j_1<…<j_k=n, \; k=1,2,…,n }
*

The points *0=t_0<t_1<…<t_m=1* (or *0=j_0<j_1<…<j_k=n*) that achieves the maximums is called a supreme partition (or just a partition for short).

There are two main functions that this package is all about, namely it is `pvar`

and `PvarBreakTest`

.
The main function in this package is `pvar`

.
It calculates the p-variation and the partition.
And the function `PvarBreakTest`

is one of the examples of p-variation applications.
It performs structural break test of vector `x`

that exams whether there are multiple
shifts in mean inside vector `x`

.

All other functions are loaded only for supporting and illustrating purposes.

### Author(s)

Author and Maintainer: Vygantas Butkus <Vygantas.Butkus@gmail.com>.

Special thanks to Rimas Norvaisa the supervisor of my studies.

### References

[1] R. M. Dudley, R. Norvaisa. An Introduction to p-variation and Young Integrals, Cambridge, Mass., 1998.

[2] R. M. Dudley, R. Norvaisa. Differentiability of Six Operators on Nonsmooth Functions and p-Variation, Springer Berlin Heidelberg, Print ISBN 978-3-540-65975-4, Lecture Notes in Mathematics Vol. 1703, 1999.

[3] R. Norvaisa, A. Rackauskas. Convergence in law of partial sum processes in p-variation norm. Lth. Math. J., 2008., Vol. 48, No. 2, 212-227.

[4] J. Qian. The p-variation of Partial Sum Processes and the Empirical Process. The Annals of Probability, 1998, Vol. 26, No. 3, 1370-1383.

### See Also

The main function is `pvar`

- it finds p-variation and the partition that maximizes `Sum_p`

function.

Other important functions is `PvarBreakTest`

it performs structural break test of vector `x`

by calculating p-variations of `BridgeT(x)`

(see `BridgeT`

).