probability

For each of the following sequences, determine the value to which it converges in probability.

(a) Let X1,X2,… be independent continuous random variables, each uniformly distributed between −1 and 1.

Let Ui=X1+X2+⋯+Xii,i=1,2,….
What value does the sequence Ui converge to in probability?

Let Wi=max(X1,X2,…,Xi),i=1,2,….
What value does the sequence Wi converge to in probability?

Let Vi=X1⋅X2⋅⋯⋅Xi,i=1,2,….
What value does the sequence Vi converge to in probability?

(b) Let X1,X2,… be independent identically distributed random variables with E[Xi]=2 and var(Xi)=9, and let Yi=Xi/2i.

What value does the sequence Yi converge to in probability?

Let An=1n∑i=1nYi. What value does the sequence An converge to in probability?

Let Zi=13Xi+23Xi+1 for i=1,2,…, and let Mn=1n∑i=1nZi for n=1,2,….
What value does the sequence Mn converge to in probability?

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