Convergence and Random Variables

Question

With the Sample space being [0,1], suppose we are looking at a sequence of random variables X_n defined by X^ω_n = ωⁿ . How can we prove that X_n converges to 0 in probability? And also what is its convergence in L²? If there even is one?

 

 

Answer 1

Showing convergence in probability:

 

For ω in [0,1), X_n(ω)=ωⁿ →0 as n→∞. Thus, except at ω=1, X_n  → X=0. Assuming the measure is a Lebsegue measure, we have almost sure convergence of  X_n to 0.  This implies, the convergence X_n of to 0 in probability. 

 

Showing convergence in L² 

Assuming further that the sequence of random variables are uniform random variables, take X= 0, and consider

lim E(X_n(ω)-x)² = lim E(ω²ⁿ) = lim ∫_[0,1]ω²ⁿdω=lim (1⁄(2n +1))=0

 

Thus, we have again an L² convergence of X_n to X=0