What is 'correct' query to fetch a cumulative sum in MySQL?

I've a table where I keep information about files, one column list contains the size of the files in bytes. (the actual files are kept on disk somewhere)

I would like to get the cumulative file size like this:

+------------+---------+--------+----------------+
| fileInfoId | groupId | size   | cumulativeSize |
+------------+---------+--------+----------------+
|          1 |       1 | 522120 |         522120 |
|          2 |       2 | 316042 |         316042 |
|          4 |       2 | 711084 |        1027126 |
|          5 |       2 | 697002 |        1724128 |
|          6 |       2 | 663425 |        2387553 |
|          7 |       2 | 739553 |        3127106 |
|          8 |       2 | 700938 |        3828044 |
|          9 |       2 | 695614 |        4523658 |
|         10 |       2 | 744204 |        5267862 |
|         11 |       2 | 609022 |        5876884 |
|        ... |     ... |    ... |            ... |
+------------+---------+--------+----------------+
20000 rows in set (19.2161 sec.)

Right now, I use the following query to get the above results

SELECT
  a.fileInfoId
, a.groupId
, a.size
, SUM(b.size) AS cumulativeSize
FROM fileInfo AS a
LEFT JOIN fileInfo AS b USING(groupId)
WHERE a.fileInfoId >= b.fileInfoId
GROUP BY a.fileInfoId
ORDER BY a.groupId, a.fileInfoId

My solution is however, extremely slow. (around 19 seconds without cache).

Explain gives the following execution details

+----+--------------+-------+-------+-------------------+-----------+---------+----------------+-------+-------------+
| id | select_type  | table | type  | possible_keys     | key       | key_len | ref            | rows  | Extra       |
+----+--------------+-------+-------+-------------------+-----------+---------+----------------+-------+-------------+
|  1 | SIMPLE       |     a | index |
sql mysql query-optimization
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