$X$, $Z_n$, $Y_n$ are independent random variables, where X is integrable, $Y_n$ has Bernoulli distribution $b(1,n^{-2})$ , $Z_n$ has Poisson disribution with parameter $n^2$. I need to check convergence of $V_n = Y_nZ_n + (1-Y_n)X$ to $X$ in mean, probability and a.s.
I can't prove that it is convergent in probability or not. It seems to me it is not.
Observe that for $\epsilon>0$: $$\{|V_n-X|>\epsilon\}\subseteq\{V_n\neq X\}\subseteq\{Y_n\neq 0\}$$so that:$$\mathsf P(|V_n-X|>\epsilon)\leq\mathsf P(V_n\neq X)\leq\mathsf P(Y_n\neq 0)=n^{-2}$$