MySQL 8.0

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CREATE TABLE log_table (
  user_id VARCHAR(5),
  date_time DATETIME,
  event_name VARCHAR(10),
  trivial int
);

INSERT INTO log_table (user_id, date_time, event_name, trivial)
VALUES
  ('001', '2020-12-10 10:00:02', 'c', 3),
  ('001', '2020-12-10 10:00:01', 'b', 9),
  ('001', '2020-12-10 10:00:40', 'e', 2),
  ('001', '2020-12-10 10:00:20', 'd', 6),
  ('001', '2020-12-10 10:00:00', 'a', 1),
  ('002', '2020-12-10 10:00:10', 'C', 9),
  ('002', '2020-12-10 10:00:50', 'D', 0),
  ('002', '2020-12-10 10:00:02', 'A', 2),
  ('002', '2020-12-10 10:00:09', 'B', 4);

Records: 9  Duplicates: 0  Warnings: 0


select * from log_table


user_id
date_time
event_name
trivial


001
2020-12-10 10:00:02
c
3

001
2020-12-10 10:00:01
b
9

001
2020-12-10 10:00:40
e
2

001
2020-12-10 10:00:20
d
6

001
2020-12-10 10:00:00
a
1

002
2020-12-10 10:00:10
C
9

002
2020-12-10 10:00:50
D
0

002
2020-12-10 10:00:02
A
2

002
2020-12-10 10:00:09
B
4


with l as (
      select l.*,
             cast(row_number() over (partition by user_id order by date_time) as signed) as seqnum
      from log_table l
     )
select l.*,
       (select group_concat(l2.event_name order by l2.date_time separator '')
        from l l2
        where l2.user_id = l.user_id and
              l2.seqnum between l.seqnum - 2 and l.seqnum + 2
       ) as new_event_name
from l


user_id
date_time
event_name
trivial
seqnum
new_event_name


001
2020-12-10 10:00:00
a
1
1
abc

001
2020-12-10 10:00:01
b
9
2
abcd

001
2020-12-10 10:00:02
c
3
3
abcde

001
2020-12-10 10:00:20
d
6
4
bcde

001
2020-12-10 10:00:40
e
2
5
cde

002
2020-12-10 10:00:02
A
2
1
ABC

002
2020-12-10 10:00:09
B
4
2
ABCD

002
2020-12-10 10:00:10
C
9
3
ABCD

002
2020-12-10 10:00:50
D
0
4
BCD


with l as (
      select l.*, full_concat,
             cast(row_number() over (partition by user_id order by date_time) as signed) as seqnum
      from log_table l join
           (select user_id,  group_concat(event_name order by date_time separator '') as full_concat
            from log_table l
            group by user_id
           ) ll
           using (user_id)
     )
select l.*, substr(full_concat, greatest(seqnum - 2, 1), least(5, seqnum + 2))
from l


user_id
date_time
event_name
trivial
full_concat
seqnum
substr(full_concat, greatest(seqnum - 2, 1), least(5, seqnum + 2))


001
2020-12-10 10:00:00
a
1
abcde
1
abc

001
2020-12-10 10:00:01
b
9
abcde
2
abcd

001
2020-12-10 10:00:02
c
3
abcde
3
abcde

001
2020-12-10 10:00:20
d
6
abcde
4
bcde

001
2020-12-10 10:00:40
e
2
abcde
5
cde

002
2020-12-10 10:00:02
A
2
ABCD
1
ABC

002
2020-12-10 10:00:09
B
4
ABCD
2
ABCD

002
2020-12-10 10:00:10
C
9
ABCD
3
ABCD

002
2020-12-10 10:00:50
D
0
ABCD
4
BCD

user_id	date_time	event_name	trivial
001	2020-12-10 10:00:02	c	3
001	2020-12-10 10:00:01	b	9
001	2020-12-10 10:00:40	e	2
001	2020-12-10 10:00:20	d	6
001	2020-12-10 10:00:00	a	1
002	2020-12-10 10:00:10	C	9
002	2020-12-10 10:00:50	D	0
002	2020-12-10 10:00:02	A	2
002	2020-12-10 10:00:09	B	4

user_id	date_time	event_name	trivial	seqnum	new_event_name
001	2020-12-10 10:00:00	a	1	1	abc
001	2020-12-10 10:00:01	b	9	2	abcd
001	2020-12-10 10:00:02	c	3	3	abcde
001	2020-12-10 10:00:20	d	6	4	bcde
001	2020-12-10 10:00:40	e	2	5	cde
002	2020-12-10 10:00:02	A	2	1	ABC
002	2020-12-10 10:00:09	B	4	2	ABCD
002	2020-12-10 10:00:10	C	9	3	ABCD
002	2020-12-10 10:00:50	D	0	4	BCD