Let un and vn converge in probability to c and d, Basic Statistics

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Let un and vn converge in probability to c and d

Let Un and Vn converge in probability to c and d, respectively. Prove the following.

(a) The sum Un + Vn converges in probability to c + d.

Hint: Show that Pr (IUn + vn - c - di > £) Pr (IUn - cl + Ivn - di £) Pr (IUn - cl > £/2 or Ivn - di £/2) Pr (IUn - cl ".?= £/2) + Pr (I vn - d i > £/2).

(b) The product Un V,, converges in probabili ty to ed.

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Basic Statistics: Let un and vn converge in probability to c and d

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Basic Statistics: Let un and vn converge in probability to c and d

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