27H (Probability and Measure)

Lévy's continuity theorem states that if characteristic functions

ϕ_{n}

ϕ

continuous at zero, then

ϕ

is a characteristic function and the corresponding laws converge weakly to its law. Conversely, convergence in distribution implies pointwise convergence of characteristic functions.

b

For

X_{j} \sim Unif [- j, j]

ϕ_{X_{j}} (t) = \frac{1}{2 j} \int_{- j}^{j} e^{i t x} d x = \frac{s i n ( j t )}{j t} .

Independence therefore gives

ϕ_{n^{- 3/2} S_{n}} (ξ) = j = 1 \prod n \frac{sin ( j ξ / n ^{3/2} )}{j ξ / n ^{3/2}} = \frac{n ^{3 n /2}}{ξ ^{n} n !} j = 1 \prod n sin (\frac{j ξ}{n ^{3/2}}) .

Uniformly for the arguments involved,

lo g \frac{s i n x}{x} = - \frac{x ^{2}}{6} + O (x^{4}) .

Hence

lo g ϕ_{n^{- 3/2} S_{n}} (ξ) = - \frac{ξ ^{2}}{6 n ^{3}} \sum_{j = 1}^{n} j^{2} + O (\frac{ξ ^{4}}{n ^{6}} \sum_{j = 1}^{n} j^{4}) ⟶ - \frac{ξ ^{2}}{18},

because

n^{- 3} \sum j^{2} \to 1/3

and the error is

O (n^{- 1})

. The limiting characteristic function is that of

N (0, 1/9)

. Lévy's theorem gives

n^{- 3/2} S_{n} \Rightarrow N (0, 1/9) .

Since

Var (X_{j}) = j^{2} /3

V_{n} := \sum_{j = 1}^{n} σ_{j}^{2} = \frac{1}{3} \sum_{j = 1}^{n} j^{2} \sim \frac{n ^{3}}{9} .

Thus

V_{n}^{- 1/2} S_{n}

differs by a factor tending to

3

from

n^{- 3/2} S_{n}

. Slutsky's theorem and part (c) yield

V_{n}^{- 1/2} S_{n} \Rightarrow N (0, 1) .

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