题目642 · 概率
Martingale Diagnosis Counterexample 2
M_n = X_(n+1)-p for iid Bernoulli(p) variables with natural filtration F_n. Is (M_n) a martingale?
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中文题目题目646 · 概率
Predictable-Transform Martingale 1
M_n = sum_(k=1)^n S_(k-1) * X_k, where X_k are iid symmetric ±1 and S_(k-1)=X_1+...+X_(k-1). Is (M_n) a martingale with respect to the natural filtration?
打开 →题目647 · 概率
Predictable-Transform Martingale 2
M_n = sum_(k=1)^n (1+S_(k-1)^2) * X_k, where X_k are iid symmetric ±1. Is (M_n) a martingale with respect to the natural filtration?
打开 →题目648 · 概率
Predictable-Transform Martingale 3
M_n = sum_(k=1)^n 1{S_(k-1)>0} * X_k for a symmetric ±1 walk. Is (M_n) a martingale with respect to the natural filtration?
打开 →题目649 · 概率
Predictable-Transform Martingale 4
M_n = sum_(k=1)^n (2+(-1)^k) * X_k with iid symmetric ±1 X_k. Is (M_n) a martingale with respect to the natural filtration?
打开 →题目631 · 概率
Terminal-Variable Projection 1
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = 1{X_1+X_2+X_3 >= 2} and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目632 · 概率
Terminal-Variable Projection 2
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = 1{X_1+X_2+X_3+X_4 = 0} and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目633 · 概率
Terminal-Variable Projection 3
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = 1{max(X_1,X_2,X_3) = 1} and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目634 · 概率
Terminal-Variable Projection 4
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = X_1+X_2+X_3+X_4 and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目635 · 概率
Terminal-Variable Projection 5
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = (X_1+X_2+X_3)^2 and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目2919 · 概率
The Standard Branching Martingale
Let $m>0$ be the offspring mean in a Galton-Watson process. Show that \[ M_n=\frac{Z_n}{m^n} \] is a martingale with respect to the natural filtration.
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