Suppose that Yn is known to converge in distribution to Po(λ) as n-> ∞. We also know that E[Xn] = E[Yn] for all values of n. Can we then assert that Xn converges in distribution to Po(λ)?
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No. We may even have $X_n=EY_n=\lambda$ for all $n$ in which case $X_n \to \lambda$.
Kavi Rama Murthy
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What if we specify that both Xn is a summation of terms A1 + A2 + … An, where each Ai has a non-zero probability of being a particular integer value. We then specify that Yn is a summation of terms B1 + B2 + … + Bn, where each Bi has a non-zero probability of being a particular integer value. – Todd Davidson Dec 10 '19 at 09:25
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@ToddDavidson EX_n=EY_N$ is too weak a condition to say anything more about the distribution functions. – Kavi Rama Murthy Dec 10 '19 at 09:28