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Let $X$ be a Possion random variable with parameter $\lambda \theta$. Show that $X-\lambda \theta \over \lambda$ converges to zero in probability, as $\lambda \rightarrow \infty$, in other words $${X-\lambda \theta \over \lambda} \rightarrow ^P 0 , as \lambda \rightarrow\infty$$ note, $\rightarrow ^P$ refer as converges in probability.

I manage to prove this via Chebyshev's inequality, but I am not very happy about the tedious computation of its expectation value. So, any other ways to prove this?

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Divide by theta, not lambda? –  Lost1 Jan 12 at 10:56
@Lost1 oops.... sorry i corrected it –  Dylan Zhu Jan 12 at 11:01
Show that the characteristic function of $Y$ converges to $1$ pointwise. Conclude that $Y$ converges to $0$ in distribution and hence also in probability. –  Stefan Hansen Jan 12 at 11:03
Then can this be done by wlln, in fact it converges a.s.? Approximate lambda by near by integers and show error is small –  Lost1 Jan 12 at 11:03

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