Consider the stochastic process (Sn)n20, such that So 0 and for n ≥ 1 Sn = 1 X₁, where (X;)21 is a collection of i.i.d random variables which take the values 0,1 and have P(X, = 1) = p > 0. = (a) Show that (Sn- np)n21 is a martingale with respect to the filtration associated with (Sn)n21. (b) Prove that IP(|S, - np| ≥t) ≤ 2e-.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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0 and for n 1 S =
Consider the stochastic process (Sn)n20, such that So
E- Xj, where (X;);21 is a collection of i.i.d random variables which take the values
0,1 and have P(X; = 1) = p> 0.
(a) Show that (S, – np)n2i is a martingale with respect to the filtration associated
with (Sn)n21.
(b) Prove that P(|S, – np| 2 t) < 2e-.
Transcribed Image Text:0 and for n 1 S = Consider the stochastic process (Sn)n20, such that So E- Xj, where (X;);21 is a collection of i.i.d random variables which take the values 0,1 and have P(X; = 1) = p> 0. (a) Show that (S, – np)n2i is a martingale with respect to the filtration associated with (Sn)n21. (b) Prove that P(|S, – np| 2 t) < 2e-.
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