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By Dimits, Krommes

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10) is not a stochastic process evaluated at time t. 3). 3. Identifying the Limit Process We should recognize that we have arrived at another critical point. Another important intellectual step is needed here. We not only must identify the limit process; we need to realize that there indeed should be a limit process. The appropriate limit process turns out to be a Brownian motion (BM). Brownian motion stochastic processes can be characterized as the real-valued stochastic processes with stationary and independent increments having continuous sample paths.

3) holds in the space D with one of the Skorohod nonuniform topologies, where cn and S are arbitrary. If P (range(S) = 0) = 0 , then plot({Sk − mk : 0 ≤ k ≤ n}) ⇒ plot(S) . Note that the functions sup, inf, range and plot depend on more than one value x(t) of the function x; they depend on the function over an initial segment. 3. 16). 16). Formulating the stochastic-process limits in D means that we can obtain many more limits for related quantities of interest, because many more quantitities of interest can be represented as images of continuous functions on the space of stochastic-process sample paths.

Note that the six observed values in each case are consistent with these pairs. g. see the papers by Erd¨ os and Kac (1946), Donsker (1951), Prohorov (1956) and Skorohod (1956). 1. Fixed space scaling. In our plots, we have let the plotter automatically determine the units on the vertical axis. 2 show that there is striking statistical regularity associated with automatic plotting. However, for comparison, it is often desirable to have common units. Interestingly, Donsker’s FCLT and the analysis of the range above shows how to determine appropriate units for the vertical axis for the centered random walk, before the simulations are run.