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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1answer
1k views

Formula for the variance of a renewal process

Let $N(t)$ be a renewal process, with a sequence of IID inter-arrival times $X_{1}, X_{2}, \dots$ having finite second moment: $EX_{i}^{2} < \infty$. How would I show that $$\mathrm{Var}N(t)= 2 \...
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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
43 views

Renewal for Levy Processes

Suppose $X(t)$ is a Levy process with almost surely positive increments (for all $t_1 < t_2$ $P(X(t_1) < X(t_2)) = 1$) Define $$\nu X(t) := \sup \{\tau \in \mathbb{R_+}| X(\tau) < t\}$$ It ...
3
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1answer
974 views

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 ...
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,...
2
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1answer
204 views

Renewal process problem

Cars, of random length $L$, arrive at a gate. The first car parks against the gate. The other arriving cars park behind at a distance uniformly distributed on $[0,1]$. Let $N(t)$ be the number of cars ...
2
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1answer
205 views

moments of Renewal function

Given a renewal process ${X_t}$. How to prove that $\lim_{t\rightarrow \infty}{E[\left(N(t)/t\right)^2]}<\infty$? Does one also have $\lim_{t\rightarrow \infty}{E[\left(N(t)/t\right)^4]}<\infty$ ...
2
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1answer
204 views

Kullback-Leibler divergence between two Markov Renewal Processes

Consider the renewal process $P(x)$ is generated according to interrenewal distribution $p(x)$, and renewal process $Q(x)$ is generated according to interrenewal distribution $q(x)$. Calculate the ...
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18 views

Lifetime expectation in renewal process

Let $X_1, X_2, \ldots$ be interarrival times like the time between two buses coming to a bus station. Let $Y(t)$ be the time until the next renewal, like the time until the next bus from a given time $...
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84 views

Renewal Limit Theorem - Proof by Krengel

I'm actually working through the Renewal Limit Theorem Proof from "Einführung in die Wahrscheinlichkeitstheorie und Statistik" by Ulrich Krengel and having problems to understand the first step, ...
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150 views

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(...
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150 views

Two independent renewal processes

We have two urns (blue and red) that are connected, and two particles, $p_1$ and $p_2$, are traveling between these urns independently. The amount of time $Z_1$ that $p_1$ spends in blue urn is iid ...
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1answer
267 views

Poisson Process (Renewal) Question

I am having difficulty with the following problem. I tried conditioning on T_n but I am unsure how to proceed with that conditional expectation. Thanks for the help!
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1answer
160 views

Renewal Reward Process Problem

People arrive at a college admissions office at rate 1 per minute. When k people have arrive a tour starts. Student tour guides are paid $20 for each tour they conduct. The college estimates that it ...
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1answer
1k views

Renewal process with geometric interarrival times

How would I go about determining the renewal function, $m(n)$, for a general $n$, if the interarrival times, $X_i$ are geometrically distributed with $P(X_i = k) = p \cdot (1-p)^{k-1}$. I believe I ...
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20 views

Renewal process: escess of life and age of the process independent.

Let $X_1,X_2,...$ be non-negative iid continuous random variables. Let $W_n=X_1+\cdots+X_n$, with $W_0=0$, and $N(t)=\max\{n\geq 0:W_n\leq t\}$. We define for this renewal process the excess of life $$...
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Is the renewal function really a distribution function?

I often read that the renewal function $R(t)=\mathbb{E}(N_t)$ is a distribution function. But shouldn't a distribution function (CDF) be smaller than $1$?
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38 views

Linking Markov Chain with Renewal Process

GIVEN: $X_0,X_1,...$ irreducible, recurrent Markov chain with transition matrix $P$ Starting state $X_0=x$ $g(m)=P\{X_m=y\}$ for some fixed state $y$ I know that the renewal process is $g(m)=b(m)+\...
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55 views

Use renewal theory for approximation.

I am confused about how to use renewal theory finding approximations required in the following questions. The questions are: A fair 4-sided die has sides labelled 1,2,3,4. Let $Y_n$ equal the ...
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37 views

Probability of short sojourns in an alternating renewal process?

Let $X_t$ be an alternating renewal process on the state space {'off', 'on'}. We can think of the process as sojourning in those states. Let $[0,\tau]$ be an interval of time and $s < \tau$. What ...
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316 views

Renewal Process with Poisson lifetimes

I'm given the following question; Consider a renewal process for which the lifetimes X1, X2, · · · are discrete r.v. having the Poisson distribution with mean λ. And then asked to find the ...
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142 views

Find the long-run reward rate using renewal theory

Given the transition probability matrix with the order of column = row = $(1,2,3)$: $$P =\pmatrix{0.2 & 0.8 & 0\\ 0.4 & 0 & 0.6\\ 1 & 0 & 0}$$ representing a DTMC. Suppose ...
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74 views

Equality involving expected value of renewal process.

Let $N_t$ be a renewal process and $B_t=t-S_{N_t}$ with $S_{N_t}=X_1+...+X_{N_t}$. If $F$ is the distribution function of $X_1$, show that $$\int_0^tE(B_t\mid X_1=x)F(dx)=\int_0^tE(B_{t-x})F(dx)$$ My ...
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42 views

renewal function and solution(exercice)

Let $N_t$ a renewal process and $U(t)=E(N_t)$ Let $B_t=S_{N_t}-t$ with $S_{N_t}=X_1+\cdots+X_{N_t}$ Show $\displaystyle U(t)=\frac{t} {E(X_1)}+\frac{E(B_t)}{E(X_1)}$. I tried: $$\frac{t} {E(X_1)}+\...
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33 views

renewal process and Markov property

Let $A_t=t-S_{N_t -1}$ with $N_t$ a renewal process 1)Show $A_t$ checks the Markov property my proof: $S_{N_t}=X_1+\cdots+X_{N_t}$ and the increments are independents $$P(S_{N_t-1}=t-y\mid S_{N_{u_1}...
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1answer
144 views

Renewal process - sample space

My question is related to the renewal process, as defined in this document. Renewal process is an arrival process in which the interarrival intervals are positive, independent and identically ...
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1answer
117 views

Renewal process large time behaviour

I have the following question: I have two types of battery, $1$ and $2$. Suppose the lifetime of $1$ is uniformly distributed on the interval $(0,3)$, battery $2$ uniformly distributed on the ...
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0answers
76 views

How to prove exchangeability for a renewal process of inter-arrival times

By definition we have that $X_1, \ldots , X_n $ are exchangeable if $X_{i_1}, \ldots, X_{i_n}$ has the same joint distribution as $X_1, \ldots , X_n $ whenever $i_1, \ldots,i_n$ is a permutation of $...
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2answers
27 views

Conditional Expectation in a Renewal Process

My professor wrote the following statement down as a starting point for a problem that I'm working on, and I don't quite see where it's coming from. We are told that $\{N(t); t \ge 0\}$ is a simple ...
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1answer
169 views

renewal process question

The question and answer below is related to the renewal process. I'm curious how the "I" changed to "H(T)" as indicated by the yellow boxes. Thanks for spending the time to look over my newbie ...
0
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1answer
70 views

renewal equation for $\sim U(0,1)$ interarrivals

renewal equation for $\sim U(0,1)$ interarrivals should be $m(t)=t+\int_0^t{m(t-s)f(s)ds}$ how can this be solved? can I make substitution $y=t-s$ to get $m(t)=t+\int_0^t{m(y)f(y)dy}$ if all I ...
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1answer
99 views

Waiting time of two independent processes

Suppose that we have two independent alternating renewal processes such that both alternate between states "0" and "1" independently. The amount of time each of them is in state "1" and state "0" ...
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2answers
122 views

Joint distribution of consecutive renewal times

Consider a discrete analog to the Poisson process. Let the sequence $X_i$ be independent geometrically (with parameter $p$) distributed random variables that signify the inter arrival times of events. ...
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0answers
9 views

How does this problem relate to renewal processes?

You plan an car trip in which half the trip (measured in distance) will be driving on highways and the other half will be driving in the city. If your car’s estimated fuel economy is 40 m.p.g on the ...
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2answers
80 views

Conditional probability of a renewal process.

Let $S_n$ denote the time of the $n$th event of the renewal process $\{N(t),t\geq0\}$ having interarrival distribution F. I need to find $P(N(t)=n|S_n=y).$ Clearly, if $y$ is the time at which the $...
0
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1answer
377 views

Expected Residual life in Renewal process with gamma interarrival distribution

I am working on problem 3.16 in Sheldon Ross's Stochastic Processes book. The problem is, "Consider a renewal process whose interarrival distribution is the gamma distribution with parameters $(n,\...
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76 views

renewal processes

Let $X_1, X_2,... $ be a discrete renewal process, in which $X_i$ denotes the time in between renewals with distribution: $Pr(X_i=1)=p$ and $Pr(X_i=2)=q=1-p. $ I want to show that the renewal ...
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0answers
53 views

Renewal Processes (Rolling a Fair Die)

What is the expected number of rolls until you see a 6 immediately followed by a 2? Let $D_i$ = number observed on the $i^{th}$ roll Assume $T_0=0$ Now let $T_1$ = min{$i\ge2$:$D_{i-1}=6$, $D_i=2$}...
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1answer
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 ...
0
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1answer
36 views

Calculate Expectation of points in a homogenous poission process with parameter $\alpha $ as a renewal process?

If a poisson process $N $ on $[0, \infty ) $ has rate $\alpha $ (ie $E N(A)=\alpha m(A) $, $m $ lebesgue measure ) can its points be represented as occurences in a renewal process with interarrival ...