Proof-of-Activity (a lottery which is using randomness from PoW output to select winners among UTXO holders) was first proposed (AFAIK) by Bentov et al. in with a goal to reward for running the nodes in the following paper https://allquantor.at/blockchainbib/pdf/bentov2014proof.pdf . Conceptually, PoA is close to PoS (e.g. Nxt is about also about a lottery which is using PoS chain to obtain randomness, see http://chepurnoy.org/blog/2014/10/inside-a-proof-of-stake-cryptocurrency-part-2/ )

However, instead of tweaking a consensus protocol, we can reward people running nodes via a smart contract. Let me describe the contract in general.

For random seed we can use id of the last block header (but in execution context 10 last blocks available, so the seed could be e.g. XOR of last few block ids).

Then lottery box contract contains a target value. A reward could be withdrawn from the lottery box if a pseudo-random hit is less than the target. Hit value of a box can be calculated as e.g. blake2b256(header.parentId || box.id) * box.value .

The contract readjusts the target with aim to have a hit every N blocks.

The lottery box then has the following conditions:

input index of the box is e.g. 0, hit box input e.g. 1

output index of the updated lottery box e.g. 0

outs[0].R4 = HEIGHT

newTarget = ins[0].R5 * (outs[0].R4 - ins[0].R4) / 10

outs[0].R5 = newTarget

blake2b256(header.parentId || box.id) < newTarget * box.value

proper amount is taken from the lotter box (it could be fixed amount or proportional to time elapsed from previous hit)

The tricky question is whether to require the box to be spent (like in the example above) or not (in this case the box would be provided as a data input). In the former case wallet must be open all the time (which is not the most secure option) while in the latter one the participation in the lottery becomes outsourceable.

Such a lottery could be used for airdropping tokens to people running the nodes.