Always null recurrence at the boundary between positive recurrence and transience?

Question

I have the following theorem:

Let $\rho$ be the traffic intensity.

a) If $\rho<1$, then $X$ is positive recurrent.

b) If $\rho>1$, then $X$ is transient.

c) If $\rho=1$, then $X$ is null-reсurrent.

It is not important what exactly $\rho$ and $X$ are. Let's assume I have proved a) and b). Then it remains to prove the case $\rho=1$:

Can I say the following: $X$ is transient if and only if $\rho>1$, and is positive recurrent if and only if $\rho<1$. If $\rho=1$ then $X$ has no choice but null-recurrent.

Is that correct?

Answer 1

a) If $\rho<1$, then $X$ is positive recurrent.

Let's assume I have proved a)

Can I say the following: $X$ is positive recurrent if and only if $\rho<1$

No, you cannot - you have only proved that $\rho <1$ is a sufficient condition for the positive recurrence of $X$, but you have not established its necessity. So you can say, $X$ is positive recurrent if $\rho<1$, but you can't use if and only if.

Let us say, in one universe there are boys, girls and wardrobes, and only three names are used: John, Jane and Woody. After spending some time there, you mention that John is only used for boys and Jane is only used for girls. You still can't conclude the Woody is only used for wardrobes, as there may happen to be a boy called Woody.

Answer 2

Is that correct?

No, first because there is no proof here, second because in some settings the boundary case is positive recurrent, respectively transient.