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\phi(x) = f(x)   \lambda \int_a^b K(x,t)\,\phi(t)\,dt
&& b_j \oplus b_i^{-1} \\
&& = (a_0 \oplus \cdots \oplus a_j) \oplus (a_0 \oplus \cdots \oplus a_i)^{-1} \\
&& = (a_0 \oplus \cdots \oplus a_j) \oplus (a_i^{-1} \oplus \cdots \oplus a_0^{-1}) \\
&& = (a_0 \oplus a_0^{-1}) \oplus \cdots \oplus (a_i \oplus a_i^{-1}) \oplus (a_{i 1} \oplus \cdots \oplus a_j) \\
&& = a_{i 1} \oplus \cdots \oplus a_j
X_N^{(2)} &=& \frac{X_0}{N}R_1 R_2 \cdots R_N   \frac{X_0}{N} R_2 R_3 \cdots R_N   \cdots   \frac{X_0}{N}R_N \\
          &=& \frac{X_0}{N} (R_1 \cdots R_N   R_2 \cdots R_N   \cdots   R_N) \\
          &=& \frac{X_0}{N} \sum_{i=1}^N \prod_{j=i}^N R_j \\
          &\equiv& \frac{X_0}{N} \sum_{i=1}^N {\cal R}_i
{\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