# Records for the moving average of a time series

## TL;DR

This paper studies how taking moving averages of i.i.d. random sequences affects the statistics of records and extremes, revealing a robust dichotomy based on distribution tail behavior.

## Contribution

It provides an asymptotic analysis and exact results showing the impact of moving averages on record statistics for different distribution types.

## Key findings

- Superexponential distributions' record statistics are asymptotically unaffected by moving averages.
- Subexponential distributions experience increased late-time record-breaking probability, scaled by a universal factor.
- The results hold across various distributions and window widths, highlighting a universal dichotomy.

## Abstract

We investigate how the statistics of extremes and records is affected when taking the moving average over a window of width $p$ of a sequence of independent, identically distributed random variables. An asymptotic analysis of the general case, corroborated by exact results for three distributions (exponential, uniform, power-law with unit exponent), evidences a very robust dichotomy, irrespective of the window width, between superexponential and subexponential distributions. For superexponential distributions the statistics of records is asymptotically unchanged by taking the moving average, up to interesting distribution-dependent corrections to scaling. For subexponential distributions the probability of record breaking at late times is increased by a universal factor $R_p$, depending only on the window width.