Converting two short-time-correlated signals with powerlaw PDF into one long-time-correlated signal

Assume a stationary, non-negative signal u(t)>=0, exponentially autocorrelated with characteristic time constant \tau_c

C_{uu}(\tau)=\left< \left(u(0)-\overline{u}\left) \left(u(\tau)-\overline{u}\left) \right> = \sigma_u^2 \;e^{-\tau/\tau_c},

having a powerlaw-shaped probability density function (PDF) with exponent a in the range [2,3], up to some upper cutoff value u_max:

P(u) \propto \theta(u_{max}\!-\!u)\;u^{-a}.

The cutoff is necessary for a finite variance of the PDF.

Assume a second random signal d(t), independent or u(t), with the same statistical properties.

From u(t) and d(t) we generate a new signal y(t) in the following way:

y(t)=\int_0^t \left[u(t^{\prime})-d(t^{\prime})\right] dt^{\prime} = \int_0^t \left[x(t^{\prime})] dt^{\prime}.

Note that x(t) is a random signal with positive and negative values, symetrically distributed around zero, with powerlaw tails. What is the autocorrelation function of y(t) ?

To build up some intuition, we can approximately replace x(t) by a simpler function z(t) which is piecewise constant over periods of lengths \tau_c:

z(t)=\sum_n \theta(t-n\tau_c) \;\theta((n\!+\!1)\tau_c-t)\;x_n,

where the heights of the steps are defined by

x_n = x(n\tau_c).

When this z(t) is integrated (instead of x(t)), the output signal y(t) will increase linearly within each step. At the end of each step, i.e. at discrete times

t_n=n \tau_c,

z(t) has performed a step of width

\Delta y_n=x_n\tau_c,

which is, of course, powerlaw distributed. In other words, embedded into the random process y(t) there is a random walk with constant waiting times and powerlaw-distributed step lengths - a Levy flight:

P(\Delta y) \propto \Delta y^{-a}.

It is known that such a Levy flight has a mean squared deviation (MSD) of

\left< \left( y(t)-y(0) \right)^2\right> \propto t^{2/(a-1)}.

So, if a=3, then the MSD of y(t) behaves diffusive (Brownian walk). If a=2, then the y-MSD behaves ballistic. For values in between this range, one obtaines a signal y(t) with fractional, superdiffusive behaviour. This, in turn, means that y(t) is long-time-correlated.

The above was only a hand-waving argument, which needs to be tested.

But if it turnes out to be true, it provides a simple biochemical mechanism for producing a long-time correlated fiber growth process: As an input, one only needs a reaction network that produces fluctuating concentrations u(t)/d(t) of proteins U/D, with statistical properties as the u(t) above (We actually know examples of such networks !).

The protein U acts as an enzyme for catalyzing the assembly of the stress fibers S from a huge reservoire of raw material R. Protein D acts a a disassembler:

R + U ---> (R-1) + S + U
(R-1) + S + D ---> R + D

So the momentary growth rate of the fiber will be proportional to x(t). Then, the signal y(t) above can be interpreted as the fiber size. We have already proposed a similar mechanism in Long Beach, California, 2008 as our "Model 3".

Maybe it is possible to replace the fluctuating disassembly process by one occuring at constant rate, such as described in post IMCM[3].