 Solution

ID: past-exam-of-the-graph-theory-and-probabilistic-combinatorics-course-of-the-university-of-oxford/2009/b/4/a/ii/solution

The assumptions imply

Var (X_{n}) = E X_{n}^{2} - μ_{n}^{2} = o (μ_{n}^{2}) .

Eventually

μ_{n} > 0

, and

X_{n} = 0

implies

∣ X_{n} - μ_{n} ∣ \geq μ_{n}

P (X_{n} = 0) \leq \frac{Var ( X _{n} )}{μ _{n}^{2}} ⟶ 0 .

No independence between the different

X_{n}

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