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Yagoda

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## Homework Statement

I want to show that if I have a consistent sequence of estimators [itex]W_n[/itex] for [itex]\theta[/itex], i.e. [itex]\lim_{n \rightarrow \infty} P(|W_n - \theta| < \epsilon) = 1[/itex], then [itex]U_n = a_nW_n + b_n[/itex] is also a consistent sequence of estimators for [itex]\theta[/itex] where [itex]\lim_{n \rightarrow \infty}a_n = 1 [/itex] and [itex]\lim_{n \rightarrow \infty}b_n = 0. [/itex]## Homework Equations

## The Attempt at a Solution

We are looking at [itex]\lim_{n \rightarrow \infty} P(|a_nW_n + b_n - \theta| < \epsilon)[/itex], which is equivalent to [itex]\lim_{n \rightarrow \infty} P(-\epsilon < a_nW_n + b_n - \theta < \epsilon)[/itex] or [itex]\lim_{n \rightarrow \infty} P(-\epsilon - b_n< a_nW_n - \theta < \epsilon - b_n) [/itex].

My question is how I can apply what I know about the limits of [itex]W_n, a_n, b_n[/itex] in this expression. While I would like to be able to say that since the a

_{n}'s go to 1 and the b

_{n}'s to 0 I can apply the consistency of Wn, but I don't know if that is acceptable.