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 Incoming links: Characteristic function of an isotropic Gaussian vector

For

d = 1

and

X \sim N (0, σ^{2})

φ_{X} (u) = e^{- σ^{2} u^{2} /2} .

σ^{2} I_{d}

, the coordinates are uncorrelated and hence independent by independence of uncorrelated jointly normal variables. Therefore

φ_{X} (u) = \prod_{j = 1}^{d} E [e^{i u_{j} X_{j}}] = \prod_{j = 1}^{d} e^{- σ^{2} u_{j}^{2} /2} = exp (- \frac{σ ^{2}}{2} j = 1 \sum d u_{j}^{2}) .