When does the following sequences of random variables follow SLLN?

69 Views Asked by At

Find the values of the constant $c$ in each case so that the Strong Law of Large Numbers holds for each sequence of independent random variables $\{X_n\}$:

(i) $P(X_n=\pm c^n)=\left(\dfrac{2}{3}\right)^{n+1},\space P(X_n=0)=1-2\left(\dfrac{2}{3}\right)^{n+1}$

(ii) $P(X_n=\pm n^c)=\dfrac{1}{2n},\space P(X_n=0)=1-\dfrac{1}{n}$

I don't have any idea on what to do because I only know how to do for iid r.v.