Convergence in probability or convergence in distribution?

Let $X$ be a random variable. let \begin{align*} Y=\alpha_1+\alpha_2 X \end{align*} where $\alpha_1$ and $\alpha_2$ are parameters.

Now let \begin{align*} Z=\hat{\alpha}_1+\hat{\alpha}_2 X \end{align*} where $\hat{\alpha}_1$ and $\hat{\alpha}_2$ are estimates of parameters.

As $n \rightarrow \infty$ , $\hat{\alpha}_1 \overset{p}{\to} \alpha_1$ and $\hat{\alpha}_2 \overset{p}{\to} \alpha_2$.

Now can I say $Y\overset{p}{\to}Z$ ($Y$is becoming $Z$ asymptotically, someone critize me by saying that it is strange to say a variable converge in probability to a variable) or $Y\overset{d}{\to} Z$ (in distribution)?

• Actually, $Z \to Y$ in probability. This of course implies convergence in distribution. The critic was because your notation is not so precise. Actually you should write $Z_n=\hat{a}_1^n+\hat{a}_2^nX$ or something like that... – Jimmy R. Nov 12 '15 at 8:09