Just do some bounds checking and wrap the index in code.
The example code below can be made more efficient, but it's written like this for clarity (sort of). Also there may be some minor mistake since I'm typing into StackOverflow rather than compiling it.
int array[10] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
int averages[10] = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0};
for (int i = 0; i < 10; i++)
{
int a = i;
int b = i + 1 > 9 ? i - 10 + 1;
int c = i + 2 > 9 ? i - 10 + 2;
int count = array[a] + array[b] + array[c];
int average = count / 3; // note this will truncate any decimal part
averages[i] = average;
}