Big jump principle for heavy-tailed random walks with correlated increments

Since its first formulation in 1964, the so-called “big jump principle” (BJP) of extreme value statistics has played an important role in fields such as large deviation theory and financial mathematics. For heavy-tailed distributions, the principle considers N independent and identically distributed (IID) random variables, and links the statistics of the sum of these variables and of their maximum value. The principle states that the probability the sum exceeds some extreme value varies in direct proportion to the probably that the single maximum value exceeds the same value. In words, large values of the sum result from large values of the maximum of the individual variables – these being the “big jumps.” Other summands play a negligible role.

In physics terms, the BJP connects a large excursion of a random walk model  with the maximum increment of its component steps. In this, the principle explains the emergence of macroscopic extreme events to the occurrence of microscopic extreme events. This scenario is very different from processes modelled with narrow distributions, where the extreme of the sum is controlled by the accumulation of many small jumps all adding up – in a generally exponentially unlikely way — to produce an extreme event.

But if the original BJP was developed for independent random variables, it remains a challenge to incorporate correlations among the variables into the theory. This is unfortunate, given the frequent importance of correlated random variables in statistical physics and many other areas. In a recent pre-print, LML External Fellow Eli Barkai and Marc Höll offer an extension of the BJP to physical systems with correlations, exploring an exactly solvable model showing the detailed influence of the correlations on the big jump principle. The model involves a random walk with correlated increments, these constructed as a weighted sum based on earlier increments and involving a positive memory kernel. Again, they assume a heavy-tailed distribution, but now include  correlations among the variables. As the authors note, this situation arises in many settings, with application in animal telemetry data, biological movements, external force fields and diffusion in porous media diffusion. The problem can also be seen as the discrete Ornstein-Uhlenbeck process with heavy-tailed noise.

Barkai and Höll offer two distinct ways to extend the BJP – that is, to derive a relationship between statistics of large values of the sum of variables and of the maximum of the variables. In either case, the main difference between the extended and “classic” BJP is that, in the case of correlations, the occurrence of large values of the sum is linked not only to one “big jump,” but also to subsequent variables following the big jump. Roughly speaking, the presence of correlations means the first big jump trails aftershocks which also influence the statistics of the extreme value of the sum variable.

The article is available at https://arxiv.org/pdf/2106.14222.pdf

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