Average Shape of a Fluctuation: Universality in Excursions of Stochastic Processes

We study the average shape of a fluctuation of a time series $x(t)$, which is the average value $⟨x(t)-x(0){⟩}_T$ before $x(t)$ first returns at time $T$ to its initial value $x(0)$. For large classes of stochastic processes, we find that a scaling law of the form $⟨x(t)-x(0){⟩}_T=T^{\alpha }f(t/T)$ is obeyed. The scaling function $f(s)$ is, to a large extent, independent of the details of the single increment distribution, while it encodes relevant statistical information on the presence and nature of temporal correlations in the process. We discuss the relevance of these results for Barkhausen noise in magnetic systems.

Figure 1

Schematic illustration of the average excursion. Top: the signal $x(t)$ is shown and two successive positive fluctuations above $x(0)$ are indicated. Bottom: the two fluctuations are rescaled and compared to the average shape.

Figure 2

Main: Normalized average fluctuation $⟨x(t){⟩}_T$ for a random walk with power-law distributed steps with $\mu =0.7$. The solid line represents the universal function (4). Circles are for several values of $T$ ranging from 199 to 9699. Inset: Normalization factor $N(T)$, which scales as $T^{1/\mu }$, indicating that $\alpha =1/\mu$. The dashed line is proportional to $T^{1/\mu }$.

Figure 3

Average excursion for a random walk in a parabolic well [Eq. (7)] for $1/\lambda =20$. From top to bottom, lines are for $T=10$, 25, 50, 100, and 250. Notice that the shape flattens to the constant value $\sqrt{4/(\pi \lambda )}$ (thin line).

Figure 4

Main: Normalized average fluctuation $⟨x(t){⟩}_T$ for a random walk fed by noise with short time memory ($\tau =1000$). Empty symbols are for uniformly distributed $\eta$; the solid ones are for power-law distributed $\eta$ with $\mu =3/2$. Circles are for a time of the order of $\tau$; triangles are for $T\gg \tau$. The solid line represents the universal function (4) after normalization. Inset: Tails for small $t/T$. The straight dashed line has slope 1.

Figure 5

Main: Normalized average fluctuation $⟨x(t){⟩}_T$ for the random accelerated particle. Triangles are for uniformly distributed $\eta$; squares and circles are for power-law distributed $\eta$, respectively, with $\mu =1.5$ and $\mu =0.7$. $T$ ranges from 1000 to 100 000. Inset: Tails for small $t/T$. The straight line has slope $3/2$.