A Graduate Course in Probability by Howard G. Tucker

Author:Howard G. Tucker
Language: eng
Format: epub, pdf
Publisher: Dover Publications

The inequality holds if we take supremum of both sides over all x. We now take lim sup of both sides as n → ∞ and use Lemma 1 to obtain

and by the arbitrariness and countability of the values of r we obtain the conclusion.

It should be noticed that the above proof was accomplished without ever verifying whether is a random variable or not.

EXERCISES

1. Prove that as defined above is a random variable.

2. Let {Xn} be a sequence of independent, identically distributed random variables with common distribution function F. Let 0 < p < 1 and x be such that F(x − ) < p < F(x + ) for all > 0. Let Xn, 1, , Xn, n, be defined by Xn, j as the jth smallest of {X1, , Xn} ; that is, Xn, 1(ω) = min {X1(ω), , Xn(ω)}, Xn, 2(ω) is the next to smallest of {X1(ω), , Xn(ω)}, etc., for every ω.

(a) Prove that Xn, 1, , Xn, n are random variables, and

(b) If [u] denotes the largest integer u, prove that Xn,[np] → x a.s.

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