Brownian meander factor for the spatial axis must be square root of the scaling factor for the time axis. The process resulting from this snip-and-scale procedure is a Brownian meander. As the name suggests, it is a piece of Brownian motion that spends all its time away from its starting point formula_5. The transition density formula_12 of Brownian meander is described as follows: For formula_13 and formula_14, and writing we have and In particular, i.e. formula_19 has the Rayleigh distribution with parameter 1, the same distribution as formula_20, where formula_21 is an exponential random variable with parameter 1. Brownian meander In the
Fractional Brownian motion Motion are independent, increments for fractional Brownian motion are not. If H > 1/2, then there is positive autocorrelation: if there is an increasing pattern in the previous steps, then it is likely that the current step will be increasing as well. If H < 1/2, the autocorrelation is negative. The process is self-similar, since in terms of probability distributions: This property is due to the fact that the covariance function is homogeneous of order 2H and can be considered as a fractal property. FBm can also be defined as the unique mean-zero Gaussian process, null at the origin, with