Introduction to Probability · Problem 9.3.3

Generating Functions — Grinstead & Snell

Mathematics — Probability · central-limit-theorem-continuous

Problem

Let

\{X_k\}

1 \leq k \leq n

, be a sequence of random variables, all with mean

\mu

and variance

\sigma^2

, and

Y_k = X^*_k

be their standardized versions. Let

S_n

and

T_n

be the sum of the

X_k

and

Y_k

, and

S^*_n

and

T^*_n

their standardized version. Show that

S^*_n = T^*_n = T_n/\sqrt{n}