Poisson tricky problem

[alamorge]

Suppose that N individuals have a certain gain "a" at t0 and that each of them at any point in time participate to an independent lottery which distribute a gain proportional to the initial gain with coefficient of proportionality b. The lottery is a Poisson with parameter L.

That is after the first lottery has been won, the distribution of gains is
a with frequency (N-1)/N
ba with frequency 1/N

Obviously at the second win it can be either of the following two cases

1.
a with frequency (N-1)/N
bba with frequency 1/N

2.
a with frequency (N-2)/N
ba with frequency 2/N

according to whether the first winner wins again or not.

After three wins things are as follows

1.
a with frequency (N-1)/N
bbba with frequency 1/N

2.
a with frequency (N-2)/N
ba with frequency 1/N
bba with frequency 1/N

3.
a with frequency (N-3)/N
ba with frequency 3/N

I am wondering which is the distribution of gains after a certain number of wins, say T, has occurred. Is that possible to recover which is its probability density function and cumulative density function?

Many thanks.