Transience of MC: Proving b>=1

[alphabeta1989]

1. Homework Statement
Consider the following model.

$X_{n+1}$ given $X_n, X_{n-1},...,X_0$ has a Poisson distribution with mean $\lambda=a+bX_n$ where $a>0,b\geq{0}$. Show that $X=(X_n)_{n\in\mathrm{N_0}}$ is a transient M.C if $b\geq 1$.2. Homework Equations

How do we approach this question? I was thinking of using the theorem below.

Let $X$ be an irreducible Markov chain with countable state space $S$. A necessary and sufficient condition for $X$ to be transient is the existence of a non-constant, non-negative super-harmonic function $\phi$.
3. The Attempt at a Solution
I was thinking of using an exponential function as a superharmonic function, but failed terribly. What superharmonic function can we use to prove transcience for $b\geq 1$ Thanks in advance.

 

[Ray Vickson]

1. Homework Statement
Consider the following model.

$X_{n+1}$ given $X_n, X_{n-1},...,X_0$ has a Poisson distribution with mean $\lambda=a+bX_n$ where $a>0,b\geq{0}$. Show that $X=(X_n)_{n\in\mathrm{N_0}}$ is a transient M.C if $b\geq 1$.


2. Homework Equations

How do we approach this question? I was thinking of using the theorem below.

Let $X$ be an irreducible Markov chain with countable state space $S$. A necessary and sufficient condition for $X$ to be transient is the existence of a non-constant, non-negative super-harmonic function $\phi$.



3. The Attempt at a Solution
I was thinking of using an exponential function as a superharmonic function, but failed terribly. What superharmonic function can we use to prove transcience for $b\geq 1$ Thanks in advance.

Something is missing: you have included no statement about what happens to/with the super-harmonic function $\phi$ in the context of $X$.

 

[alphabeta1989]

Something is missing: you have included no statement about what happens to/with the super-harmonic function $\phi$ in the context of $X$.

I attempted to use functions such as $\phi(x) = e^{bx}, e^{(b-1)x}$, but all of them are not superharmonic w.r.t $X$. What type of functions should I attempt?

 

[Ray Vickson]

I attempted to use functions such as $\phi(x) = e^{bx}, e^{(b-1)x}$, but all of them are not superharmonic w.r.t $X$. What type of functions should I attempt?

You are missing the whole point: WHAT is supposed to happen if I give you a superharmonic function? You quoted only half of a theorem; the other half is vital!

 

[alphabeta1989]

You are missing the whole point: WHAT is supposed to happen if I give you a superharmonic function? You quoted only half of a theorem; the other half is vital!

I am sorry about that! This is the definition of a superharmonic function!

Let $X$ be a time-homogeneous irreducible Markov chain with countable state space $S$ and one-step transition probability matrix $P(x, y)$. A function $\phi: S \rightarrow R$ is said to be superharmonic for X at $x \in S$ if $\sum_{y\in S} P(x,y)\phi(y)\leq\phi(x)$

 

[Ray Vickson]

I am sorry about that! This is the definition of a superharmonic function!

Let $X$ be a time-homogeneous irreducible Markov chain with countable state space $S$ and one-step transition probability matrix $P(x, y)$. A function $\phi: S \rightarrow R$ is said to be superharmonic for X at $x \in S$ if $\sum_{y\in S} P(x,y)\phi(y)\leq\phi(x)$

If you don't care about signs, just getting a superharmonic $\phi$ is easy: in this case, $\phi(x) = -x$ is superharmonic if $a>0,\: b \geq 1$. However, if you want a non-negative $\phi$ it is harder. You can follow the construction in
http://math.stackexchange.com/questions/165913/markov-chains-recurrence-and-transcience