Question

Let $\left \{ Yn \right \}_{n=1}^{\infty }$ be a sequence of random variables, and let Y be a random variable on the same sample space. Let A_n(ϵ) be the event that |Y_n − Y | > ϵ. It can be shown that a sufficient condition for Y_n to converge to Y w.p.1 as n → ∞ is that for every ϵ > 0,

$\sum_{n=1}^{\infty }P(An(\varepsilon ))< \infty$

(a) Let $\left \{ Xi \right \}_{n=1}^{\infty }$ be independent uniformly distributed random variables on [0, 1], and let Y_n = min(X₁, . . . , X_n). In class, we showed that Yn → 0 w.p.1. Prove the same result by using the sufficient condition given above.

(b) Let $\left \{ Zn \right \}_{n=1}^{\infty }$ be exponential random variables with parameter α that are not necessarily independent, and let Vn = Zn/n. Use the sufficient condition above to show that V_n → 0 w.p.1.

Answer 1

ANSWER::

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