Convergence in probability.
For each of the following sequences, determine whether it converges in probability to a constant. If it does, enter the value of the limit. If it does not, enter the number “999".
1) Let X1, X2,… be independent continuous random variables, each uniformly distributed between −1 and 1.
a) Let Ui=X1+X2+⋯+Xii, i=1,2,…. What value does the sequence Ui converge to in probability? (If it does not converge, enter the number “999". Similarly in all below.)
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b) Let Σi=X1+X2+⋯+Xi, i=1,2,…. What value does the sequence Σi converge to in probability?
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c) Let Ii=1 if Xi≥1/2, and Ii=0, otherwise. Define,
Si=(I1+I2+⋯+Ii)/i.
What value does the sequence Si converge to, in probability?
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d) Let Wi=max{X1,…,Xi}, i=1,2,…. What value does the sequence Wi converge to in probability?
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e) Let Vi=X1⋅X2⋯Xi, i=1,2,…. What value does the sequence Vi converge to in probability?
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2) Let X1, X2,…, be independent identically distributed random variables with E[Xi]=2 and Var(Xi)=9, and let Yi=Xi/2i.
a) What value does the sequence Yi converge to in probability?
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b) Let An=1n∑i=1nYi. What value does the sequence An converge to in probability?
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c) 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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