Do a research about the random walk process and its properties. Compare your finding with your applications drawing your personal conclusions. Explain based on your exercise the beaviour of the distribution of the stochastic process (check out “Donsker’s invariance principle”). What are, in particular, its mean and variance at time n ?
Random walk process
In mathematics, a random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.
Integer random walk (1,-1)
An elementary example of a random walk is the random walk on the integer number line,Z , which starts at 0 and at each step moves +1 or −1 with equal probability.
Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler: all can be approximated by random walk models, even though they may not be truly random in reality.
In the 9A2
We have used the integer random walk in -1 1 and compared it with a standard normal ditribution (both moltiplicated by a fixed factor sqrt(1/n)) and with this comparison, by the histograms that rappresent the empiricals frequencies distribution,we could see that the two graphs are quite similar both with mean 0 and variance 1, this is in line with what stated by the Donsker’s invariance principle(functional central limit theorem)
Functional central limit theorem
the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem, state that: As random variables taking values in the Skorokhod space D[0,1], the random function W^n converges in distribution to a standard Brownian motion
W:=(W(t)) with t in [0,1] and n->∞