Weak Convergence of Probability Measures

Weak Convergence of Probability Measures on ℝ^d

Among several concepts of convergence that are being used in Probability theory, the weak convergence has a special role, as it is related not to values of random variables, but to their probability distributions. In a simplest case of a sequence {X _n } of real valued random variables (or vectors with values in ℝ^d, d ≥ 1) defined on probability spaces $({\Omega }_{n},{\mathcal{F}}_{n},{P}_{n})$, we say that a sequence {X _n } converges weakly (or in law) to a random variable X if

$${ \lim \limits_{n\rightarrow +\infty }}{F}_{n}(x) = F(x)$$

(1)

for each x ∈ ℝ where the function F is continuous. Here F _n and F are distribution functions of X _n and X, respectively. The notation for this kind of convergence is X _n ⇒ X, or ${X}_{n} \mathop \rightarrow \limits^{{\mathcal{L}}}X$. The convergence defined by (1) can be as well thought of as being a convergence of corresponding distributions, i.e., probability measures defined on \(({\mathbb{R}}^{d},{\mathcal{B}}^{d})\)...