Before implementing any of the selected tests, the existence of correlations between market returns was studied, as they offer early insights into the markets that do not follow a

random walk. When a market is identified as being weak-form efficient, stock prices follow a

random walk, indicating that price changes are independent and not related to each other, and that historical information is fully and instantaneously reflected in current prices.

The benchmark models used for comparison are the consensus forecast from Blue Chip, an AR model, and a

random walk. I use Blue Chip forecasts rather than forecasts from the SPF, as the SPF does not forecast at a monthly frequency.

Simulation is done by adding

Random Walk, Random Waypoint and Gauss-Markov mobility model and also the traffic pattern scheme as the interval of the packet that will be delivered as follow [8].

Although he did not exploit all richness of his model, Langevin obtained in the long-time regime Einstein's result, namely, by <[x.sup.2]> = 2([k.sub.B]T/6[pi]a[mu])t, where a is the radius of particle, [mu] is fluid viscosity, T is temperature, and [k.sub.B] is the Boltzmann's constant, hence bridging Brownian motion,

random walk, and diffusion, a view soon quantified experimentally by Perrin [18].

Scher, "Application of continuous time

random walk theory to tracer test measurements in fractured and heterogeneous porous media," Groundwater, vol.

Kesten and Spitzer [1] proved that when the

random walk and the random scenery belong to the domains of attraction of different stable laws of indices 1 < [alpha] [less than or equal to] 2 and 0 < [beta] [less than or equal to] 2, respectively, then there exists [delta] > 1/2 such that {[n.sup.-[delta]][K.sub.[??]n[??]], t [greater than or equal to] 0} converges weakly as n [right arrow] [infinity] to a continuous [delta]-self-similar process with stationary increments, [delta] being related to [alpha] and [beta] by [delta] = 1 - [[alpha].sup.-1] + [([alpha][beta]).sup.-1].

Gonzelez, "The

random walk of an electrostatic field using parallel infinite charged planes," Revista Mexicana de Fisica, vol.

By using (6), it is easy to prove that the unique stationary distribution of the simple

random walk becomes

The MLE method is generally used to compute the amount of white noise, flicker noise and

random walk noise in the time series (Zhang et al., 1997; Langbein and Johnson, 1997; Mao et al., 1999; Williams et al., 2004; Langbein and Bock, 2004).

When comparing the sea stars' movements to

random walk models, we used the sea star's positions from every 25th frame.

Specifically, when the

random walk reaches a vertex v, that there are three options for the walker: (a) continue the

random walk to the neighbors of v, (b) abandon the

random walk or (c) stop the walk and emit a confidence according to prior knowledge.

Given an RGG G(n,R), the cover time of a

random walk is defined as the expected length that enable a source packet visiting all storage nodes in AS network at least once, and denote by [T.sub.c].

Rossi (2013) showed that it is still difficult to beat the

random walk, particularly in an out-of-sample exercise.

The auto-correlation of randomness for the chosen period rejected the

Random Walk Hypothesis (RWH) for daily and weekly index returns but documented the existence of RWH for monthly index returns.