Universal Record Statistics of Random Walks and Lévy Flights

Abstract

We compute exactly the statistics of the number of records in a discrete-time random walk model on a line where the walker stays at a given position with a nonzero probability 0 ≤ p ≤ 1, while with the complementary probability 1 − p, it jumps to a new position with a jump length drawn from a continuous and symmetric distribution f 0 (η). We show that, for arbitrary p, the statistics of records up to step N is completely universal, i.e., independent of f 0 (η) for any N .We also compute the connected two-time correlation function C p (m 1 , m 2 ) of the record-breaking events at times m 1 and m 2 and show it is also universal for all p. Moreover, we demonstrate that C p (m 1 , m 2 ) < C 0 (m 1 , m 2 ) for all p, indicating that a nonzero p induces additional anti-correlations between record events. We further show that these anti-correlations lead to a drastic reduction in the fluctuations of the record numbers with increasing p. This is manifest in the Fano factor, i.e. the ratio of the variance and the mean of the record number, which we compute explicitly. We also show that an interesting scaling limit emerges when p → 1, N → ∞ with the product t = (1 − p) N fixed. We compute exactly the associated universal scaling functions for the mean, variance and the Fano factor of the number of records in this scaling limit.

Record statistics of financial time series and geometric random walks

“…for large N , is proportional to √ N and the mean record age r ∝ N [7]. These results have further been generalized to the case of random walk with a constant drift with application to stock market data [15] and also to multiple random walkers [11].…”

“…Significantly, it is independent of the value of N . In contrast to quantities like the mean number of records which depend on N [7,12,15], the distribution of record ages is characteristic statistical property of record breaking events independent of the length of data. Further, as N increases, the range over which the power law is valid also increases implying that the tail behavior is a finite size effect.…”

“…Record age distribution will provide insights into how long a record can be expected to survive and is useful in hazard estimation problems. Though the mean record age has been analytically determined for random walk problems in earlier works [7,15], there have been no results for the distribution of record age.…”

“…Recently, the record statistics of correlated series such as the positions of random walker was studied [7,10,11]. Random walk is a fundamental model in physics and has applications in many areas including the dynamics of stock markets.…”

“…In physics, records statistics is useful in understanding the behavior of stochastic motion of a domain wall in metallic ferromagnetic materials [5] and as an alternative indicator of quantum chaos in kicked rotor model [6]. Even as the record breaking events continue to enjoy media attention, there is also an increased research interest in the statistical study of record events [7][8][9][10][11][12].…”