Convolve a minimum phase wavelet with delta peak let's assume I have two time series: one with a delta peak at position $x$ and another time series with a minimum phase wavelet that starts at time $y$ with a maximum peak at position $z$.

Now, when I convolve both time series, is the beginning of my minimum-phase wavelet at position $x$ or is the maximum amplitude of the minimum-phase wavelet at position $x$ ?

I know, that when I convolve a zero-phase wavelet, the maximum amplitude is at the position of the delta function.

I am puzzled..........
If I understand the question properly. Consider the case when the delta is at time zero. In this case the convolution does not do anything to the other signal i.e. it starts at time y and the maximum peak will be at time z.

Now, if we delay the delta to time x, then the output of the convolution will also be delayed by x. So the signal will start at time x+y, and the maximum peak will be at time x+z.

If I haven't understood correctly, you may need to include some figures to more clearly explain the situation.