# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $$P(S_{N_t-1}=t-y\mid S_{N_{u_1}... 0answers 53 views ### Renewal Processes (Rolling a Fair Die) What is the expected number of rolls until you see a 6 immediately followed by a 2? Let D_i = number observed on the i^{th} roll Assume T_0=0 Now let T_1 = min{i\ge2:D_{i-1}=6, D_i=2}... 0answers 42 views ### renewal function and solution(exercice) Let N_t a renewal process and U(t)=E(N_t) Let B_t=S_{N_t}-t with S_{N_t}=X_1+\cdots+X_{N_t} Show \displaystyle U(t)=\frac{t} {E(X_1)}+\frac{E(B_t)}{E(X_1)}. I tried:$$\frac{t} {E(X_1)}+\...
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 $\lim_{t\rightarrow \infty}{E[\left(N(t)/t\right)^4]}<\infty$ ...