Generating random numbers from Alpha-stable pdf which expends in time (in a loop)

I asked first on another site, decided to ask here because I go no answers.

The following line of Matlab code generates 100000 random numbers from an Alpha-stable pdf (here Alpha=0.5):

Rand = random('Stable',0.5,0,1,0,[1,100000]);

The distribution of Rand exactly matches the the 'theoretical' curve generated by

PDF = makedist('Stable','alpha',0.5,'beta',0,'gam',1,'delta',0);

x = -5:.1:5;

PDF = pdf(PDF,x);

figure

plot(x,PDF,'r-.');

(to check, use, e.g.:)

Data=Rand;

Middle=0.01;

PosBinsUP=10.^(log10(Middle):0.05:log10(max(abs(Data))));

PosBinsDown=10.^(log10(Middle):0.05:log10(abs(min(Data))));

xbins=[-flip(PosBinsDown) -Middle:0.2:Middle PosBinsUP];

[xpdf ypdf]= plotpdfc(Data, xbins);

plot(xpdf(1:end-1),ypdf(1:end-1),'Or');

xlim([-5,5]);

My question is:

How do I generate these Rands in a loop over 't', such that their distribution will expand in time, namely P(Rand) = t^(-1/Alpha) W(Rand/ t^(1/Alpha))?

[Jeff Miller]

It isn't clear what you are trying to do. Is the idea to use a different value of alpha for each t, and then generate one random value with that alpha, something like this?

     for t=1:1000
         alpha = f(t);  % not sure what function you have in mind
         Rand(t) = random('Stable',alpha,0,1,0,[1,1]);  
     end

[Erez]

@Jeff Miller Not at all, I want to change the scaling of the PDF, so that tit expend in time. I think that I got it - should I make 'gamma' in random('stable',alpha,beta,gamma,delta,[1,1]) be 1/sqrt(Alpha)? Based on the example of the Gaussian here: https://www.mathworks.com/help/stats/stable-distribution.html , where gamma=1/sqrt(2) ?

Or maybe nother scaling? The sqrt(2) is becase x~t^(1/2). In a Levy process x~t^(1/Alpha). Am I correct about the interpretation of how to do it in Matlab?

[Erez]

@ Jeff Miller I still don't see how ths could would make my PDF change with time. I want that different iterations of the loop over t will yields a broader PDF, corresponding to the sum of IID random numbers of where the number of summands is growing, and the Levy-generalized central limit theorem.

Of course, I could actually just sum random numbers, to get the PDF. But I am looking for a more direct way to obtain them, using the Matlab function above.

Thanks!

[Jeff Miller]

Sorry, I don't really understand this distribution, so I don't have anything useful to contribute. It just seems that if you want a random number from the stable distribution at each t, then your only option is to adjust the stable's parameter values to be whatever you want for each t.