Question: Law of large numbers for random variables without expectation. Let .Xi /i1 be i.i.d. real-valued random variables having no expectation, i.e., Xi 62 L1. Let

Law of large numbers for random variables without expectation. Let .Xi /i1 be i.i.d.

real-valued random variables having no expectation, i.e., Xi 62 L1. Let a 2 N be arbitrary.

Show the following:

(a) P.jXnj > an infinitely often/ D 1. Hint: Use Problem 4.5.

(b) For the sums Sn D Pn iD1 Xi we have P.jSnj > an infinitely often/ D 1 and thus lim supn!1 jSnj=nD1almost surely.

(c) If all Xi  0, we even obtain Sn=n!1almost surely.

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