In words, the random walk has independent increments and those increments have a limiting normal distribution, with a variance reflecting the size of the interval (b-a) over which the increment is taken It should not be surprising, therefore, that the limit of the sequence of function r( constructed from the random walk preserves these properties in the limit n an appropriate sense. In fact, these properties form the basis of the definition of the Wiener process Definition Let(S, F, p) be a complete probability space. Then W: S[0, 1]Ris a standard Wiener process if each of r E [ 0, 1],w(, r) is F-measurable, and in addition (1). The process starts at zero: PW(, 0)=0=1 (2). The increments are independent:if0≤ao≤a1….≤ak≤1,then w(, ai)-W(, ai-1)is independent of w(, ai)-w(, ai-1),j=1,.,k ,j+ (3). The increments are normally distributed: For 0<a<b<1, the increment r(, b)-w(, a) is distribut In the definition, we have written W(, a) explicitness; whenever convenient however, we will write W(a) instead of w(, a), analogous to our notation else- where. The Wiener process is also called a brownian motion in honor of nor bert Wiener(1924), who provided the mathematical foundation for the theory of random motions observed and described by nineteenth century botanist Robert Brown in 1827 2.2 Functional central limit Theorems We earlier defined convergence in law for random variables, and now we need to extend the definition to cover random functions. Let s( represent a continuous- time stochastic process with S(r)representing its value at some date r for r E 0,1. Suppose, further, that any given realization, S() is a continuous function of r with probability 1. For Sr(T_1 a sequence of such continuous functionIn words, the random walk has independent increments and those increments have a limiting normal distribution, with a variance reflecting the size of the interval (b − a) over which the increment is taken. It should not be surprising, therefore, that the limit of the sequence of function WT (·) constructed from the random walk preserves these properties in the limit in an appropriate sense. In fact, these properties form the basis of the definition of the Wiener process. Definition: Let (S, F,P) be a complete probability space. Then W : S × [0, 1] → R 1 is a standard Wiener process if each of r ∈ [0, 1], W(·,r) is F-measurable, and in addition, (1). The process starts at zero: P[W(·, 0) = 0] = 1. (2). The increments are independent: if 0 ≤ a0 ≤ a1... ≤ ak ≤ 1, then W(·,ai) − W(·,ai−1) is independent of W(·,aj ) − W(·,aj−1), j = 1,..,k, j 6= i for all i = 1,...,k. (3). The increments are normally distributed: For 0 ≤ a ≤ b ≤ 1, the increment W(·,b) − W(·,a) is distributed as N(0,b − a). In the definition, we have written W(·,a) explicitness; whenever convenient, however, we will write W(a) instead of W(·,a), analogous to our notation else￾where. The Wiener process is also called a Brownian motion in honor of Nor￾bert Wiener (1924), who provided the mathematical foundation for the theory of random motions observed and described by nineteenth century botanist Robert Brown in 1827. 2.2 Functional Central Limit Theorems We earlier defined convergence in law for random variables, and now we need to extend the definition to cover random functions. Let S(·) represent a continuous￾time stochastic process with S(r) representing its value at some date r for r ∈ [0, 1]. Suppose, further, that any given realization, S(·) is a continuous function of r with probability 1. For {ST (·)} ∞ T =1 a sequence of such continuous function, 6