When are analyst bonuses paid?

"paid" is not the same as "earned"

Like at my company, bonuses are earned in the calendar year, calculated/determined (based on pre-defined metrics and YE results) in February, and paid in March. You have to be working for the company on that day in March to get the payment. This is fairly standard, but obviously, every firm has their own schedule/cadence, so don't look for a rule of thumb to be that universally applicable. 

If you are trying to "time" the hiring process, it may not matter much, as many first year bonuses are simply prorated for first year. As in get hired on July 1, and get 50% of the applicable bonus when paid net year. You may be able to negotiate the right to a full bonus (i.e. not prorated by hire date) as a form of "sign on" bonus (but paid next cycle), depending on the firm this may or may not be a slick way to negotiate to get more pay without causing any headaches (it's a point worth asking, to be fair).

There can also be policies like... All hired before July 1 get full bonus, those hired after October 1 get zero bonus, or new analysts get a fixed bonus, etc. I've seen variants like that before, and yes, you can sometimes negotiate them. 

Timing your start date is to have little effect, but generally, the longer you work there, the more you have rights to in the next bonus cycle (similar process with annual pay bumps, COLAs, PTO, etc. could be at play).