Maximum difference of Poisson process

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I am trying to understand this remark in a paper by Bollobás and Riordan:




Let $X_1, X_2,dots$ be the points of a Poisson process on $[0,
infty]$ with rate $m$, so, setting $X_0 = 0,$ the variables $X_i −
X_i−1$ are iid exponentials with mean $1/m.$ Let $Y_i =
sqrtX_mi,$ and let $D_m = maxY_i − Y_i−1,1 leq i <
infty$, noting that this maximum exists with probability one.




How do you show the "maximum exists with probability one"?



(Clarification: $m$ is an arbitrary natural number, and $X_mi$ means the $(mi)^textth$ point, where $mi=mtext times i$ (in particular, $X_mi$ is not "just another name" for $X_i$).)



Bollobás, B., & Riordan, O. M. (2003). Mathematical results on scale-free random graphs. Handbook of graphs and networks: from the genome to the internet, 1-34.




probability-theory stochastic-processes poisson-process




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    up vote
    3
    down vote

    favorite
    1












    I am trying to understand this remark in a paper by Bollobás and Riordan:




    Let $X_1, X_2,dots$ be the points of a Poisson process on $[0,
    infty]$ with rate $m$, so, setting $X_0 = 0,$ the variables $X_i −
    X_i−1$ are iid exponentials with mean $1/m.$ Let $Y_i =
    sqrtX_mi,$ and let $D_m = maxY_i − Y_i−1,1 leq i <
    infty$, noting that this maximum exists with probability one.




    How do you show the "maximum exists with probability one"?



    (Clarification: $m$ is an arbitrary natural number, and $X_mi$ means the $(mi)^textth$ point, where $mi=mtext times i$ (in particular, $X_mi$ is not "just another name" for $X_i$).)



    Bollobás, B., & Riordan, O. M. (2003). Mathematical results on scale-free random graphs. Handbook of graphs and networks: from the genome to the internet, 1-34.




    probability-theory stochastic-processes poisson-process




    share|cite|improve this question
























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      I am trying to understand this remark in a paper by Bollobás and Riordan:




      Let $X_1, X_2,dots$ be the points of a Poisson process on $[0,
      infty]$ with rate $m$, so, setting $X_0 = 0,$ the variables $X_i −
      X_i−1$ are iid exponentials with mean $1/m.$ Let $Y_i =
      sqrtX_mi,$ and let $D_m = maxY_i − Y_i−1,1 leq i <
      infty$, noting that this maximum exists with probability one.




      How do you show the "maximum exists with probability one"?



      (Clarification: $m$ is an arbitrary natural number, and $X_mi$ means the $(mi)^textth$ point, where $mi=mtext times i$ (in particular, $X_mi$ is not "just another name" for $X_i$).)



      Bollobás, B., & Riordan, O. M. (2003). Mathematical results on scale-free random graphs. Handbook of graphs and networks: from the genome to the internet, 1-34.




      probability-theory stochastic-processes poisson-process




      share|cite|improve this question














      I am trying to understand this remark in a paper by Bollobás and Riordan:




      Let $X_1, X_2,dots$ be the points of a Poisson process on $[0,
      infty]$ with rate $m$, so, setting $X_0 = 0,$ the variables $X_i −
      X_i−1$ are iid exponentials with mean $1/m.$ Let $Y_i =
      sqrtX_mi,$ and let $D_m = maxY_i − Y_i−1,1 leq i <
      infty$, noting that this maximum exists with probability one.




      How do you show the "maximum exists with probability one"?



      (Clarification: $m$ is an arbitrary natural number, and $X_mi$ means the $(mi)^textth$ point, where $mi=mtext times i$ (in particular, $X_mi$ is not "just another name" for $X_i$).)



      Bollobás, B., & Riordan, O. M. (2003). Mathematical results on scale-free random graphs. Handbook of graphs and networks: from the genome to the internet, 1-34.





      probability-theory stochastic-processes poisson-process





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      edited Aug 18 at 21:42

























      asked Aug 10 at 23:04









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