As I understand your question, we need a recursive query here:

with 
    data as (
        select p.*, row_number() over(partition by personid order by fromdate) rn
        from @payment p
    ),
    rcte as (
        select d.*, todate as refdate from data d where rn = 1
        union all
        select d.*, 
            case when datediff(d, r.refdate, d.fromdate) < 30 then r.refdate else d.todate end
        from rcte r
        inner join data d on d.personid = r.personid and d.rn = r.rn + 1
    )
select * from rcte where refdate = todate order by personid, fromdate

The first CTE enumerates the payments of each person. Then, the recursive query starts from the first payments, and processes rows in order, while keeping track of the "reference" start date and changing it when its gap with the current record exceeds 30 days. The last step is to just filter on the "first" record per group.

fiddle

If a person may have more than 100 payments, then you need option (maxrecursion 0) at the end of your query.

with cte as (
  select PersonId , fromdate , todate,
       lag(todate) over (partition by PersonId order by fromdate) as lag_todate
 from Payment 
),
cte2 as (
  select  *,
        datediff(d, lag_todate, fromdate ) DaysSinceLastindex,
        MIN(case when datediff(d, lag_todate, fromdate) >=30 then todate end) 
                     over (partition by PersonId order by fromdate) as new_fromdate
  from cte
),
cte3 as (
  select *, case when new_fromdate is null
               then datediff(d, lag_todate, fromdate )
               else datediff(d, new_fromdate, fromdate ) end as datediff
  from cte2
)
select PersonId, fromdate, todate, lag_todate, DaysSinceLastindex, 
  case when GREATEST(DaysSinceLastindex, datediff) >=30 OR lag_todate is null then 'YES' end as EXPECTED
from cte3
order by PersonId, fromdate

First cte to get previous todate.

cte2 to get DaysSinceLastindex and the new fromdate to compare with if it is >=30 days.

cte3 to get datediff between current date and the starting date or the with new record if exist

Result :

PersonId    fromdate    todate      lag_todate  DaysSinceLastindex  EXPECTED
1           2014-07-05  2014-07-06  null        null                YES
1           2014-07-15  2014-07-16  2014-07-06  9                   
1           2014-09-23  2014-09-24  2014-07-16  69                  YES
1           2014-10-15  2014-10-16  2014-09-24  21                  
1           2014-10-30  2014-10-31  2014-10-16  14                  YES
2           2014-07-20  2014-07-25  null        null                YES
2           2014-08-20  2014-08-21  2014-07-25  26                  
2           2014-09-21  2014-09-22  2014-08-21  31                  YES

Demo here