Start the Markov chain at a state $i$ and define the strictly positive first return time $T_i^{+}=\inf \{n\ge 1:X_n=i\}$. A recurrent state satisfies ${\mathbb{P}}_i(T_i^{+}<\infty )=1$; a transient state has this probability strictly below one. Equivalently a recurrent state is visited infinitely often almost surely when started there, whereas a transient state receives finitely many visits almost surely. The recurrence criterion by return probabilities gives the equivalent testStates in a communicating class share this classification. In an irreducible Markov chain all states communicate, so the chain is called a recurrent Markov chain if all its states are recurrent and a transient Markov chain if all are transient.

By part (c) with $\lambda =1$, let $T_n=E_1+\cdots +E_n\sim \mathrm{Gamma}(n,1)$ for independent and identically distributed exponential random variables. The requested integral is $\mathbb{P}(T_n\le n)$. Each summand has expectation one and variance one, so the central limit theorem givesZero is a continuity point of the standard normal distribution, and consequently

An exponential distribution of rate $\lambda$ is a Gamma distribution with shape one. Repeated additivity of independent gamma distributions with a common rate therefore givesWith integer shape this is also called an Erlang distribution.

Independence makes the moment-generating function of the sum the product of the two moment-generating functions. For $t<\lambda$,The expectations are finite on, for example, $(-\lambda /2,\lambda /2)$, as is the moment-generating function of the candidate Gamma distribution. The uniqueness theorem for moment-generating functions therefore applies, provingThis is additivity of independent gamma distributions with a common rate; a common rate is essential to this specific formula.

Integrating $e^{tx}$ against the Gamma density gives the moment-generating function of a gamma distribution:The change of variables $y=(\lambda -t)x$ reduces the integral to $\Gamma (r)=(r-1)!$ using the gamma function. For $t\ge \lambda$ the defining integral is infinite. In particular the moment-generating function is finite on a neighbourhood of zero.