I have a process that is currently being effectively modelled by a Poisson Binomial distribution (wiki link). We have access to all of the constituent probability values of the independent trials; however, the number of trials can be large (500-1000+). There are also many processes modelled in this way, so imagine making any calculation based off of the above many times.
For each such process, is there an efficient way to construct a probability representing k+ successful events?
The linked wikipedia article goes into much discussion on "The probability of having k successful trials out of a total of n", but I take this to be the probability of getting "exactly k successful trials". For my case, I am interested in "k or more successes."
Is there a reasonable (ideally closed-form) efficient solution to this? Pardon also if I misunderstood the mass function section on the wiki as well. My maths background is not nearly as great as it could be, so it would be appreciated to help my understanding by breaking down any solution if possible.