- 2 Consider The Random Walk W Wn Nzo On Z Where Wn Wo X Xn And X X2 Are Independent Identically Dis 1 (113.02 KiB) Viewed 12 times
2. Consider the random walk W = (Wn)nzo on Z where Wn Wo+ X₁ + ··· + Xn, and X₁, X2,... are independent, identically distributed random variables with 3 3 1 P(Xn = -1) P(Xn 1) P(Xn 2) 8' 8' 4 We define the hitting times Tk = inf{n 20: Wnk}, where infØ = +∞0. For k, m≥ 0, let xm) be the probability that the random walk visits the origin by time m given that it starts at position k, that is, (m) Xk = P(To ≤ m | Wo = k). (0) (a) Give Xk for k0. For m≥ 1, by splitting according to the first move, show that 1 (m-1) 3 (m) 3 (m-1) (m-1) Xk +=k+1 8 k-1 8 + Xk+2 (Vk > 1) 4 and x(m) = 1. [5 marks] For k0, let k be the probability that the random walk ever visits the origin given that it starts at position k, that is, k = P(To <∞| Wo = k) (b) Prove that rm) ↑ xk as m → ∞. Deduce that 3 3 1 x1 = x₂ + X3. 8 4 [4 marks] (c) By splitting according to the value of Tk-1, show that, for k ≥ 2, ∞ P(To <∞ | Wo= k) = P(Tk-1 = i| Wo= k) P(To <∞ | Wo = k; Tk-1 = :i). i=1 Deduce that P(To <∞| Wo= k) = P(To <∞ | W₁ = 1) P(To <∞| W₁ = k − 1) [4 marks] and hence x = (x₁)k for all k ≥ 0. (d) Show that either x₁ = 1 or x₁ = 1/2. [2 marks] = = = -