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Assignment 4: Due 12noon, Mon 12th December 2022
Please submit answers to Exercises 4.7, 4.8 and 4.21. Submit a single pdf file and assign questions to
pages in gradescope before confirming your submission. Do not submit multiple images. (You can find the
MS Lens app as part of the university’s MS Office suite for smartphones/tablets, or you can use the CIS
printer-scanners to scan your handwritten work.)
You can find the questions on the problems sheet, accessible from ULTRA, repeated below:
Exercise 4.7. Let X1, X2, . . . be a sequence of independent geometric random variables with success
probability p ∈ (0, 1), i.e., Pr(X1 > k) = q
k
, where q = 1 ? p. Show that
lim sup
n→∞
Xn
log n
=
1
log(1/q)
, a.s.
Hint: Show that for every ε > 0 we have Pr(An(ε) i.o.) = 0 and Pr(An(?ε) i.o.) = 1, where
An(δ) := n
Xn >
1+δ
log(1/q)
log n
o
.
Exercise 4.8. In the setup of Exercise 4.7, let Mn = max1≤k≤n Xk. Find a constant c ∈ R such that
limn→∞
Mn
log n = c a.s.
Hint: Prove both that lim supn→∞
Mn
log n ≤ c a.s. and lim infn→∞
Mn
log n ≥ c a.s.
Exercise 4.21. Let Z be a non-negative random variable. Lyapunov’s inequality states that for any
0 < p ≤ q < ∞, (E(Z p ))1/p ≤ (E(Z q ))1/q. In this exercise you will give two different proofs of this. (a) Prove Lyapunov’s inequality by Jensen’s inequality and truncation: let Wn = (Z ∧ n) p and consider (EWn) q/p . (b) Prove Lyapunov’s inequality by an application of H¨older’s inequality for appropriately chosen X and Y .