Suppose I have a length of slab $\Delta z,$ consisting of two types of material. The process $Y(z)$ controls the type of material where the state space is $S = (y_1,y_2).$

In this small interval of slab the process switches to the other material with probability proportional to $\Delta z,$ say $\lambda\Delta z$ and remains the same with probability $1-\lambda\Delta z,$ where $\lambda$ is a positive constant so $Y(z)$ is homogeneous and symmetric in the two materials. The semigroup has the form

$$P_{\Delta z} = \begin{bmatrix} 1-\lambda\Delta z & \lambda\Delta z \\ \lambda\Delta z & 1 - \lambda\Delta z \end{bmatrix}$$ with infinitesimal generator

$$\mathcal{L} = \lambda\begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}.$$ The trajectories of this process are constructed via first introducing the increasing sequence

$$0 < Z_0 < Z_1 < Z_2 < ... < Z_n < ...$$

of successive random transition points, where the process switches material. In this model the layer sizes are

$$Z_1,Z_2 - Z_1,...,Z_n - Z_{n-1},...$$

and form a sequence of independent random variables with the common exponential distribution with parameter $\lambda$

$$\mathbb{P}[Z_n-Z_{n-1}\leq z] = 1 - e^{-\lambda z}$$

for $n\geq 1.$

I am really struggling how to see how this sequence has an exponential distribution. Could someone explain this to me or provide a reference?

  • 4
    $\begingroup$ A Markov chain must comply with the memoryless propety (Markov property). Only two kinds of distributions are memoryless: geometric distributions of non-negative integers and the exponential distributions of non-negative real numbers. Your case deals with the real numbers. $\endgroup$ Oct 4, 2021 at 13:00
  • $\begingroup$ @FeedbackLooper Thanks for clarifying that...so here $\mathbb{P}[Z_n - Z_{n+1}\leq z] = 1 - e^{-\lambda z}$ is the cumulative distribution function? $\endgroup$ Oct 4, 2021 at 14:55
  • 1
    $\begingroup$ Yes, it is the cumulative distribution for $Z_n-Z_{n+1}$. $\endgroup$ Oct 4, 2021 at 16:34


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