Record statistics for multiple random walks

Abstract

We study the statistics of the number of records R(n,N) for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance σ(2) of the jump distribution is finite and (II) when σ(2) is divergent as in the case of Lévy flights with index 0<μ<2. In both cases we find that the mean record number R(n,N) grows universally as ~α(N) sqrt[n] for large n, but with a very different behavior of the amplitude α(N) for N>1 in the two cases. We find that for large N, α(N) ≈ 2sqrt[lnN] independently of σ(2) in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, α(N) ≈ 4/sqrt[π], independently of 0<μ<2. For finite σ(2) we argue-and this is confirmed by our numerical simulations-that the full distribution of (R(n,N)/sqrt[n]-2sqrt[lnN])sqrt[lnN] converges to a Gumbel law as n → ∞ and N → ∞. In case II, our numerical simulations indicate that the distribution of R(n,N)/sqrt[n] converges, for n → ∞ and N → ∞, to a universal nontrivial distribution independently of μ. We discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the Standard & Poor's 500 index.

Records in fractal stochastic processes

“…Recently, the extreme event studies have been developed using various approaches, such as the density of states [3,4], firstpassage and return-time statistics [6][7][8], persistence [9], interoccurrence time statistics [10][11][12] and record statistics [13][14][15]. Among them, there has been considerable interest in investigating the record statistics, which has found many fruitful applications in diverse complex systems, such as spin glasses [16,17], adaptive processes [18], domain wall dynamics [19], avalanche dynamics [20], stock prices [21,22], global warming [23,24], growing network [25], high-temperature superconductors [26], the ant movements dynamics [27], flood dynamics [28], sport statistics [29,30], earthquakes [31,32], and evolutionary biology [33,34].…”

Record statistics of financial time series and geometric random walks

“…Inspite of such growing interest in correlated series, very few works have focussed on the records statistics in empirical stock data [11,12,16]. In this paper, we report on the record statistics of empirical stock data to understand two quantities of interest not studied earlier, namely, (i) the distribution of record age and (ii) the distribution of longest record ages.…”

“…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].…”

“…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].…”