Security deposit interest

[Freeman Schultz]

When a building is sold and the security deposits are passed onto the new owner, does whatever interest accrued when the security deposits were in the possession of the seller carry over to the buyer? 

For example, in New Jersey, the security deposits have to be in accounts that accrue interest that needs to be distributed to the tenants at the end of the year. Now when a new owner takes over, are the responsible for distributing the interest that was generated by the previous owner?

Does the security deposit interest carry over after the building is sold? Is the new owner responsible to distribute that interest if the tenant never received it from the previous seller?