mysql equivalent of over function in lower versions - mysql

Question

I wrote the following query for a project of mine, it is working perfect, however it is working only for mysql 8 since I am using 'over' function in the query which is not supported by earlier mysql versions. I need to rewrite the query without using over so that it will work for lower versions of mysql as well.
$new_query = "
update ospos_sales_items t
inner join (
select sales_items_id, sum(total_amount) over(order by sales_items_id) sum_rate
from ospos_sales_items t
where payment_type = 'Credit' AND t.customer_id = $pid
) t1 on t1.sales_items_id = t.sales_items_id
set
t.payment_type = 'Cash',
t.total_amount = case when t1.sum_rate >= $amount then t1.sum_rate - $amount else (t.total_amount) end,
t.payment_type = case when t1.sum_rate >= $amount then 'Credit' else 'Cash' end
where t1.sum_rate - (t.total_amount) < $amount AND t.customer_id = $pid
";
I tried the below but it is not working.
$new_query = "
update ospos_sales_items t
inner join (
select sales_items_id, sum(total_amount) sum_rate
from ospos_sales_items t
where payment_type = 'Credit' AND t.customer_id = $pid
) t1 on t1.sales_items_id = t.sales_items_id
set
t.payment_type = 'Cash',
t.total_amount = case when t1.sum_rate >= $amount then t1.sum_rate - $amount else (t.total_amount) end,
t.payment_type = case when t1.sum_rate >= $amount then 'Credit' else 'Cash' end
where t1.sum_rate - (t.total_amount) < $amount AND t.customer_id = $pid
";
UPDATE:
I have simplified the subquery that is using over so that an answer can be easy, please check the fiddle suggest how to write the query without using over function but get the same results as the fiddle
select sum(total_amount) over(order by item_id) sum_rate from mytable order by item_id

Answer 1

select sum(y.total_amount) sum_rate
from mytable x
join mytable y
on y.item_id <= x.item_id
group
by x.item_id
order
by x.item_id