# Properties of transition density functiions

1. Jul 3, 2009

### mathy_girl

I hope some of you know something about transition density functions. I'm wondering if there are some nice properties I can use...

Suppose p is a transition density function, defined as follows:

p(t,f,a|T,F,A) dF dA := Prob(F<f<F+dF, A<a<A+dA | F(t)=f, A(t)=a).

My question: what happens to p and its derivatives if we take limits of F and/or A to infinity? Is there a source where I can refer to for this?

2. Jul 4, 2009

### EnumaElish

This is a new concept for me, but I'll try if you help me understand. When you posted p(t,f,a|T,F,A) dF dA did you mean to write p(t,f,a|T,F,A, dF, dA)? It would seem to me that Lim(F+dF) = Lim(F) = infinity as F ---> infinity, and similarly for A. So my guess is p ---> 0 as both F and A approach infinity. You can see this more clearly if you define p as F<f<F+dF, A<a<A+dA, with weak inequality.

3. Jul 4, 2009

### mathy_girl

I ment to say that p is defined as the (transition) probability of t going to T, f going to F and a going to A. Multiplying with increments dA and dF is because p itself is a probability distribution, which has to be integrated (it does not have a value in a point).

Does this make it a bit more clear?

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