The objective of this problem is to know the construction of Brownian motion from a symmetric...

 

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The objective of this problem is to know the construction of Brownian motion from a symmetric random walk. Problem: Consider the random variables X, as follows. X = 8 if the kth coin toss results in heads, and X = - if the kth coin toss results in tails. Thus, 8 with probability 2 X = with probability Where 8 is the length of each subinterval corresponds to a time slot of length 8. -8 6 26 36 46 56 66 76 86 96 106 He wad 1. Determine E(X) and Var(X). 2. Define the process W(1) as follow. Let W(0)-0 and at time - 8n, the value of W() is given by W(t)=W (8n) a) Calculate E(W(t)) and Var(W(t)) for any time t (0,00). b) Apply Central Limit Theorem to show that W(t)-N(0,t) with continious smples paths X (PI) The objective of this problem is to know the construction of Brownian motion from a symmetric random walk. Problem: Consider the random variables X, as follows. X = 8 if the kth coin toss results in heads, and X = - if the kth coin toss results in tails. Thus, 8 with probability 2 X = with probability Where 8 is the length of each subinterval corresponds to a time slot of length 8. -8 6 26 36 46 56 66 76 86 96 106 He wad 1. Determine E(X) and Var(X). 2. Define the process W(1) as follow. Let W(0)-0 and at time - 8n, the value of W() is given by W(t)=W (8n) a) Calculate E(W(t)) and Var(W(t)) for any time t (0,00). b) Apply Central Limit Theorem to show that W(t)-N(0,t) with continious smples paths X (PI)