 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2013/m/2/b/solution

Put

Y_{n} = n (1 - M_{n}) \geq 0

. For fixed

x \geq 0

and sufficiently large

n > x

P (Y_{n} > x) = P (M_{n} < 1 - x / n) = (1 - x / n)^{n} ⟶ e^{- x} .

For

x < 0

its distribution function is zero. The limiting distribution function is therefore

F (x) = 0

for

x < 0

F (x) = 1 - e^{- x}

for

x \geq 0

, the exponential distribution of rate one. Since this distribution function is continuous everywhere,

n (1 - M_{n}) d Exp (1) .

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