 Solution

ID: past-exam-of-the-probability-course-of-the-university-of-oxford/2021/1/c/iii/solution

There is a stronger conclusion than the requested limit. For

x > 0

N_{n} - n \geq x n

N_{n} \geq n + 1

, because

N_{n}

is an integer. In particular, at least

n

arrivals have occurred by time

n

, so the

n

T_{n} \leq n

. This contradicts

T_{n} - n \geq y n

for

y > 0

. Thus the two events are disjoint:

P (\frac{N _{n} - n}{n} \geq x, \frac{T _{n} - n}{n} \geq y) = 0 for every n .

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