Questions tagged [renewal-processes]

Suppose $\{X_i\}_{i = 1}^\infty$ are i.i.d random variables, such that $P(X_1 > 0) = 1$. Then the corresponding renewal process is $\nu(t) = \max\{n \in \mathbb{N} | \Sigma_{i = 1}^n X_i \leq t\}$. Here $t \in \mathbb{R}_+$.

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When Superposition of Two Renewal Processes is another Renewal Process?

When superposition of two renewal processes is another renewal process? If you merge (superpose) two Poisson processes with parameters $\lambda_1$ and $\lambda_2$, the outcome is another Poisson ...
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103 views

Characterizing superposition of two renewal processes

This is a follow-up question of "When superposition of two renewal processes is another renewal process?". How can we characterize the superposition of two renewal processes? The superposition of ...
3
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1answer
428 views

Splitting a renewal process

This is a follow-up question of the question "When superposition of two renewal processes is another renewal process?". How can we split a renewal process $P$ into a renewal process $P_1$ and ...
3
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1answer
455 views

Induction proof on independent increment property

Define Poisson process is a renewal process in which the interarrival intervals have exponential distribution. $S_n$ is the arrival epoch of the $n$th arrival, $N(t)$ is the number of arrivals in $(0,...
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limiting variance in Renewal Theory

Let $\{X_n\}_{n\ge 0}$ be a renewal process and define $S_n=\sum_{i=0}^{n}X_i$ and $N(t)=\sum_{n=1}^\infty\mathbb{1}[S_n\le t]$. Let $E(X_i)=\mu$ and $\text{Var}(X_i)=\sigma^2$. Let $V(t)=\text{Var}(N(...