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Prove convergence in probability for n * Poisson variable to zero

  1. Aug 8, 2013 #1
    The problem:

    Let [itex]\mu_{n} = \frac{1}{n}[/itex] for [itex]n \in \mathbb{N}[/itex]. Let [itex]X_{n} \; \mathtt{\sim} \; \textrm{ Poisson}\left( \lambda_{n} \right)[/itex].

    Let [itex]Y_{n} = n X_{n}[/itex]. Show that [itex]Y_{n} \xrightarrow{P} 0 [/itex].

    Work I've done:

    I've shown that [itex]X_{n} \xrightarrow{P} 0[/itex] by showing that [itex]\mathbb{P} \left( \left| X_{n} \right| > \epsilon \right) \; \to \; 0[/itex] as [itex]n \to \infty [/itex]. (By getting [itex] \epsilon [/itex] and [itex] \delta [/itex] so that the limit definition is satisfied.)

    I've also tried to show convergence in quadratic mean (which did not converge to 0) and convergence in distribution (which I could not do).

    Relevant equations:

    [itex]X_{n} \xrightarrow{P} X[/itex] if for every [itex] \epsilon > 0 [/itex], [itex]\mathbb{P} \left( \left| X_{n} - X \right| > \epsilon \right) \; \to \; 0[/itex] as [itex]n \to \infty [/itex].

    If the distribution converges in quadratic mean, it converges in probability.

    If the distribution converges in distribution and it is a point distribution then it converges in probability.
  2. jcsd
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