Below is a theorem that I was able to verify via code but am not sure about how do I prove it with mathematics. If anyone can take a look at this and tell me how the mathematics behind this works it would be of great help. Thank you.
Theorem: The parameters of the Weibull distribution of a combination of a time-series of lengths $X_1, X_2, X_3...X_n$ obtained from $M$ sets of Weibull parameters is equal to the shape and scale parameter of a time series of total length N, where N is the sum of the lengths.
I used maximum likelihood estimation to determine the time series of 10 different Weibull parameters each of length 100 and combined them to get a time series of length 1000 and Weibull distribution of this 1000 time series is reasonably accurate to what I would expect. Is there any mathematical proof that anyone can help so as to support this work on.