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Ludmilka [50]
3 years ago
13

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.)
b) Let Σi=X1+X2+⋯+Xi, i=1,2,…. What value does the sequence Σi converge to in probability?
c) Let Ii=1 if Xi≥1/2, and Ii=0, otherwise. Define,
Si=I1+I2+⋯+Iii.
What value does the sequence Si converge to, in probability?
d) Let Wi=max{X1,…,Xi}, i=1,2,…. What value does the sequence Wi converge to in probability?
e) Let Vi=X1⋅X2⋯Xi, i=1,2,…. What value does the sequence Vi converge to in probability?
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?
b) Let An=1n∑i=1nYi. What value does the sequence An converge to in probability?
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?

Mathematics
1 answer:
nikklg [1K]3 years ago
6 0

Answer:

Check the explanation

Step-by-step explanation:

Kindly check the attached images below for the step by step explanation to the question above.

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Step-by-step explanation:

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where a1, a2, ... are scalars, and c1 is the vector of column 1.

Then we should write:

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Here i used the element with subindex 3 and 1, but is the same for every subindex.

Notice that if A is m*n, then [At] is n*m

Now, the row space of [At] will be, same as before.

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Where b1, b2, ... are scalars and the r's are the vector of each row.

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b1 = a1, b2 = a2, b3 = a3, ...

We will have that the row space of [At] is the same as the column space of A.

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