 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2013/m5/a/solution

M_{X} (t) = E [e^{tX}]

on the set where this expectation is finite. For independent, identically distributed summands, independence factors the expectation:

M_{S_{n}} (t) = E i = 1 \prod n e^{t X_{i}} = i = 1 \prod n M (t) = M (t)^{n} .

The identity holds wherever the factors are finite; in the remaining parts the bounded summands make them finite for every real

t

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