f = g + x_{n} h

, where

g

is independent of

x_{n}

de g g \leq k

, and

de g h \leq k - 1

. Put

m_{3} = E X_{n}^{3}

∣ m_{3} ∣ \leq C

, and independence gives

E (g + X_{n} h)^{4} \leq ∥ g ∥_{4}^{4} + 6 ∥ g ∥_{4}^{2} ∥ h ∥_{4}^{2} + 4 C ∥ g ∥_{4} ∥ h ∥_{4}^{3} + C ∥ h ∥_{4}^{4} .

Choose

A = A (C) \geq 1

so large that

\frac{6}{A} + \frac{2 C}{A ^{3/2}} \leq 2, \frac{C}{A ^{2}} + \frac{2 C}{A ^{3/2}} \leq 1.

Induction on

n

, using

2 a b^{3} \leq a^{2} b^{2} + b^{4}

and the orthogonal identity

∥ f ∥_{2}^{2} = ∥ g ∥_{2}^{2} + ∥ h ∥_{2}^{2}

, now yields

E f (X)^{4} \leq A^{2 k} (E f (X)^{2})^{2} .

E X_{i}^{3} = 0

and

C \leq 9

, the cubic term vanishes. Taking

A = 3

and applying the induction hypothesis gives exactly

E f^{4} \leq 9^{k} ∥ g ∥_{2}^{4} + 6 \cdot 3^{k} 3^{k - 1} ∥ g ∥_{2}^{2} ∥ h ∥_{2}^{2} + 9 \cdot 9^{k - 1} ∥ h ∥_{2}^{4} = 9^{k} (∥ g ∥_{2}^{2} + ∥ h ∥_{2}^{2})^{2} = 9^{k} (E f^{2})^{2} .

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