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Recently Referred Formulae

{\rm V}[X_N^{(2)}] &=& \frac{X_0^2}{N^2} \left\{ 2\sum_{j>i}\mu^{j-i}(\mu^2  \sigma^2)^{N-j 1}   \sum_{i=1}^N (\mu^2  \sigma^2)^i - \left( \sum_{i=1}^{N}\mu^i \right)^2 \right\} \\
 &\equiv& \frac{X_0^2}{N^2} \left\{ 2\sum_{j>i}\mu^{j-i}\nu^{N-j 1}   \sum_{i=1}^N \nu^i - \left( \sum_{i=1}^{N}\mu^i \right)^2 \right\}
{\rm V}[X_N^{(2)}] &=& \frac{X_0^2}{N^2} \left\{ 2\sum_{j>i} \mu^{N - j   1}\mu^{N - i   1}   \sum_{i=1}^N \mu^i -\left( \sum_{i=1}^N \mu^i \right)^2 \right\} \\
 &=& \frac{X_0^2}{N^2} \left\{ 2\sum_{j<i} \mu^{i}\mu^{j}   \sum_{i=1}^N \mu^i -\left( \sum_{i=1}^N \mu^i \right)^2 \right\} \\
 &=& 0
{\rm E}\left[ \sum_{i,j} {\cal R}_i {\cal R}_j \right] &=& 2{\rm E}\left[\sum_{j>i}{\cal R}_i {\cal R}_j \right]   {\rm E}\left[ \sum_{i=1}^N{\cal R}_i^2 \right] \\
 &=& 2 \sum_{j>i}\mu^{j-i}(\mu^2   \sigma^2)^{N-j 1}   \sum_{i=1}^N (\mu^2   \sigma^2)^i
{\rm d}y &=& \frac{1}{x}{\rm d}x - \frac{1}{2x^2} ({\rm d}x)^2\\
 &=& \frac{1}{x}(\mu x {\rm d}t   \sigma x {\rm d}W_t) - \frac{1}{2x^2}(\sigma^2 x^2 {\rm d}t)\\
 &=& (\mu - \frac{1}{2}\sigma^2){\rm d}t   \sigma {\rm d}W_t
{\rm V}[R_1 R_2 \cdots R_N] &=& (\sigma^2   \mu^2)^N - \mu^{2N} \\
 &=& \sum_{i=0}^N {}_N C_i \sigma^{2i}\mu^{2(N-i)} - \mu^{2N} \\
 &=& \sum_{i=1}^N {}_N C_i \sigma^{2i}\mu^{2(N-i)}