  We study some limit theorems for the law of a generalized one-dimensional
diffusion weighted and normalized by a non-negative function of the local time
evaluated at a parametrized family of random times (which we will call a
clock). As the clock tends to infinity, we show that the initial process
converges towards a new penalized process, which generally depends on the
chosen clock. However, unlike with deterministic clocks, no specific
assumptions are needed on the resolvent of the diffusion. We then give a path
interpretation of these penalized processes via some universal $ \sigma
$-finite measures.

Probability