Suppose we list the decimal representations of the positive even numbers from left to right. Determine the $2015^{th}$ digit in the list.
To determine the $2015^{th}$ digit in the concatenation of the decimal representations of positive even numbers, we need to break down the problem into manageable parts by considering the number of digits contributed by different ranges of even numbers.

1. **One-digit even numbers:**
   The one-digit even numbers are \(2, 4, 6, 8\). There are 4 such numbers, each contributing 1 digit.
   \[
   \text{Total digits from one-digit numbers} = 4 \times 1 = 4
   \]

2. **Two-digit even numbers:**
   The two-digit even numbers range from 10 to 98. There are 45 such numbers (since \( \frac{98 - 10}{2} + 1 = 45 \)), each contributing 2 digits.
   \[
   \text{Total digits from two-digit numbers} = 45 \times 2 = 90
   \]

3. **Three-digit even numbers:**
   The three-digit even numbers range from 100 to 998. There are 450 such numbers (since \( \frac{998 - 100}{2} + 1 = 450 \)), each contributing 3 digits.
   \[
   \text{Total digits from three-digit numbers} = 450 \times 3 = 1350
   \]

4. **Four-digit even numbers:**
   The four-digit even numbers start from 1000. We need to find how many more digits are required to reach the $2015^{th}$ digit.
   \[
   \text{Digits needed} = 2015 - (4 + 90 + 1350) = 2015 - 1444 = 571
   \]

   Each four-digit even number contributes 4 digits. To find how many four-digit even numbers are needed to contribute 571 digits:
   \[
   \frac{571}{4} = 142 \text{ remainder } 3
   \]

   This means we need 142 full four-digit even numbers and 3 additional digits from the next number.

5. **Identifying the specific number:**
   The first four-digit even number is 1000. The $142^{nd}$ four-digit even number can be found as follows:
   \[
   a_n = 1000 + 2(n-1) = 1000 + 2 \times 141 = 1000 + 282 = 1282
   \]

   The next number after 1282 is 1284. Since we need 3 more digits, we look at the first three digits of 1284, which are "128".

6. **Conclusion:**
   The $2015^{th}$ digit is the third digit of 1284, which is 8.

The final answer is \(\boxed{8}\).