For any sequence of RVS {Xn} show that -1 max X0 n= Sn 1k...

 

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For any sequence of RVS {Xn} show that -1 max X₁0 ⇒ n=¹ Sn 1≤k<n Section 7.4 Strong Law of Large Numbers ∞. (a) (5 points) Show that Problem #14Let X₁, X₂, ... be a sequence of independent RVs with E₁ var (Xk)/k² < k=1 n n var (Xk) · PO →0 n → ∞ k=1 (b) (5 points) Does the converse also hold? = ∞. Problem #15(5 points) Let {o} be a sequence of real numbers such that Σ1 Show that there exists a sequence of independent RVs {X} with var(Xk) = ok, k = 1,2,..., such that n-¹ Σk_₁[Xk - E(X)] does not converge to 0 almost surely. -1 k=1 For any sequence of RVS {Xn} show that -1 max X₁0 ⇒ n=¹ Sn 1≤k<n Section 7.4 Strong Law of Large Numbers ∞. (a) (5 points) Show that Problem #14Let X₁, X₂, ... be a sequence of independent RVs with E₁ var (Xk)/k² < k=1 n n var (Xk) · PO →0 n → ∞ k=1 (b) (5 points) Does the converse also hold? = ∞. Problem #15(5 points) Let {o} be a sequence of real numbers such that Σ1 Show that there exists a sequence of independent RVs {X} with var(Xk) = ok, k = 1,2,..., such that n-¹ Σk_₁[Xk - E(X)] does not converge to 0 almost surely. -1 k=1