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New analytical expressions for the moment of the peak to average power ratio (PAPR) of ${n}$ independent and identically distributed (i.i.d.) zero-mean complex circular Gaussian random variables (r.v.s) are derived. These expressions give the higher order moments. For an orthogonal frequency division multiplexing (OFDM) system with ${n}$ subcarriers, the ${n}$ samples in each OFDM symbol are taken as i.i.d. with each following a zero-mean complex circular Gaussian distribution. The PAPR statistics of these samples play an important role as performance metrics. An exact closed form expression for the $\ell $ th moment of the PAPR obtained in earlier work is used to derive a compact expression and a simple expression in terms of a function $H_{n}(-\ell)$ ; an alternative expression for $H_{n}(-\ell)$ is also presented. The variance obtained from the $\ell $ th moment is shown to be asymptotically equal to a previous result for a large number of r.v.s.