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Convergence in Probability

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A. If g is a real-valued function that is continuous at a, then g(a^) converges in probability to g(a).
b. Let Y1, Y2, ...Yn be independent random variables, each with probability density function for 0 < y <1, 0 elsewhere. Show that converges in probability to some constant and find the constant.

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1. Proofs
a. Prove Theorem 9.2(d): If g is a real-valued function that is continuous at , then converges in probability to .
b. Let Y1, Y2, ...Yn be independent random variables, each with probability density function for 0  y 1, 0 elsewhere. Show that converges in probability to some constant and find the constant.
Solution
a) Since is consistent for , given and , however small, we can find an such that
, whenever . (1)
Again, since g is a real-valued function that is continuous at , given , we can choose such that

Thus
[From (1)]
That is, , whenever .
Hence converges in probability to .

[Note that means and hence or equivalently ]

b) Since the p.d.f. of Y is given by
for 0  y 1, 0 elsewhere
we have
=
=
and hence =
Let .
Then

and

Since and as , converges in probability to the constant = 0.75

[Note that if is a sequence of estimators of such that and as , then is a consistent estimator of ]

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