Suppose that Xn = X + Zn n/2' where Zn is any random variable with E{Z}...

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Suppose that Xn = X + Zn n¹/2' where Zn is any random variable with E{Z} = cnº, say, with c> 0 and a E R fixed, and X is any other random variable. (a) Let e > 0. Use Chebyshev's inequality to show that P(|Xn − X| > €) ≤ C €²n¹-a (b) For what values of a does the argument in part (a) prove that Xn converges in probability to X? (c) For the values of a identified in part (b), what other mode of convergence of Xn to X is assured (without any further calculations)? Suppose that Xn = X + Zn n¹/2' where Zn is any random variable with E{Z} = cnº, say, with c> 0 and a E R fixed, and X is any other random variable. (a) Let e > 0. Use Chebyshev's inequality to show that P(|Xn − X| > €) ≤ C €²n¹-a (b) For what values of a does the argument in part (a) prove that Xn converges in probability to X? (c) For the values of a identified in part (b), what other mode of convergence of Xn to X is assured (without any further calculations)?