Scaling exponents for ordered maxima

Abstract

We study extreme value statistics of multiple sequences of random variables. For each sequence with N variables, independently drawn from the same distribution, the running maximum is defined as the largest variable to date. We compare the running maxima of m independent sequences and investigate the probability S(N) that the maxima are perfectly ordered, that is, the running maximum of the first sequence is always larger than that of the second sequence, which is always larger than the running maximum of the third sequence, and so on. The probability S(N) is universal: it does not depend on the distribution from which the random variables are drawn. For two sequences, S(N)∼N(-1/2), and in general, the decay is algebraic, S(N)∼N(-σ(m)), for large N. We analytically obtain the exponent σ(3)≅1.302931 as root of a transcendental equation. Furthermore, the exponents σ(m) grow with m, and we show that σ(m)∼m for large m.

Record statistics of a strongly correlated time series: random walks and Lévy flights

“…An interesting connection between the case of i.i.d. random variables and the case of random walks is given in [129] where the relation ν K ≈ σ K /2 is observed (numerically) to be a good approximation.…”

“…From this joint probability (141) which is fully explicit in this case, using (143) and (144), the full statistics of the record number, the age of the longest lasting record max,N or the probability of record breaking Q N can be obtained, following the lines detailed in section 3.1, and yielding the results given in equations (123), (128) and (129). This joint probability (141) should be useful to compute any observable related to the ages of the lattice random walk bridge.…”

“…Consider now K ≥ 1 such sequences. These sequences are said to be perfectly ordered if the corresponding staircases do not cross [129]. The probability of this event has a power-law decay [129]:…”

“…From this joint distribution (150), together with equations ( 112) and ( 152), it is possible to compute the statistics of all the observables related to the ages of the record of the random walk bridge with symmetric exponential jumps. In particular one obtains rather straightforwardly the results for the distribution of the number of records in (126), for the record breaking probability in (127) or for max,N in (129).…”

Maxima of two random walks: universal statistics of lead changes

“…The exponents β k (µ) do not depend on µ when µ > 2, i.e., for random walks. For ordinary random walks, the exponents β k were studied numerically in [13], while approximate values for these exponents were computed in [38].…”