Convergence of Random Variables - Properties

Properties

The chain of implications between the various notions of convergence are noted in their respective sections. They are, using the arrow notation:

\begin{matrix} \xrightarrow{L^s} & \underset{s>r\geq1}{\Rightarrow} & \xrightarrow{L^r} & & \\ & & \Downarrow & & \\ \xrightarrow{a.s.} & \Rightarrow & \xrightarrow{\ p\ } & \Rightarrow & \xrightarrow{\ d\ } \end{matrix}

These properties, together with a number of other special cases, are summarized in the following list:

  • Almost sure convergence implies convergence in probability:
    X_n\ \xrightarrow{as}\ X \quad\Rightarrow\quad X_n\ \xrightarrow{p}\ X
  • Convergence in probability implies there exists a sub-sequence which almost surely converges:
    X_n\ \xrightarrow{p}\ X \quad\Rightarrow\quad X_{k_n}\ \xrightarrow{as}\ X
  • Convergence in probability implies convergence in distribution:
    X_n\ \xrightarrow{p}\ X \quad\Rightarrow\quad X_n\ \xrightarrow{d}\ X
  • Convergence in r-th order mean implies convergence in probability:
    X_n\ \xrightarrow{L^r}\ X \quad\Rightarrow\quad X_n\ \xrightarrow{p}\ X
  • Convergence in r-th order mean implies convergence in lower order mean, assuming that both orders are greater than one:
    X_n\ \xrightarrow{L^r}\ X \quad\Rightarrow\quad X_n\ \xrightarrow{L^s}\ X, provided r ≥ s ≥ 1.
  • If Xn converges in distribution to a constant c, then Xn converges in probability to c:
    X_n\ \xrightarrow{d}\ c \quad\Rightarrow\quad X_n\ \xrightarrow{p}\ c, provided c is a constant.
  • If Xn converges in distribution to X and the difference between Xn and Yn converges in probability to zero, then Yn also converges in distribution to X:
    X_n\ \xrightarrow{d}\ X,\ \ |X_n-Y_n|\ \xrightarrow{p}\ 0\ \quad\Rightarrow\quad Y_n\ \xrightarrow{d}\ X
  • If Xn converges in distribution to X and Yn converges in distribution to a constant c, then the joint vector (Xn, Yn) converges in distribution to (X, c):
    X_n\ \xrightarrow{d}\ X,\ \ Y_n\ \xrightarrow{d}\ c\ \quad\Rightarrow\quad (X_n,Y_n)\ \xrightarrow{d}\ (X,c) provided c is a constant.
    Note that the condition that Yn converges to a constant is important, if it were to converge to a random variable Y then we wouldn’t be able to conclude that (Xn, Yn) converges to (X, Y).
  • If Xn converges in probability to X and Yn converges in probability to Y, then the joint vector (Xn, Yn) converges in probability to (X, Y):
    X_n\ \xrightarrow{p}\ X,\ \ Y_n\ \xrightarrow{p}\ Y\ \quad\Rightarrow\quad (X_n,Y_n)\ \xrightarrow{p}\ (X,Y)
  • If Xn converges in probability to X, and if P(|Xn| ≤ b) = 1 for all n and some b, then Xn converges in rth mean to X for all r ≥ 1. In other words, if Xn converges in probability to X and all random variables Xn are almost surely bounded above and below, then Xn converges to X also in any rth mean.
  • Almost sure representation. Usually, convergence in distribution does not imply convergence almost surely. However for a given sequence {Xn} which converges in distribution to X0 it is always possible to find a new probability space (Ω, F, P) and random variables {Yn, n = 0,1,…} defined on it such that Yn is equal in distribution to Xn for each n ≥ 0, and Yn converges to Y0 almost surely.
  • If for all ε > 0,
then we say that Xn converges almost completely, or almost in probability towards X. When Xn converges almost completely towards X then it also converges almost surely to X. In other words, if Xn converges in probability to X sufficiently quickly (i.e. the above sequence of tail probabilities is summable for all ε > 0), then Xn also converges almost surely to X. This is a direct implication from the Borel-Cantelli lemma.
  • If Sn is a sum of n real independent random variables:
then Sn converges almost surely if and only if Sn converges in probability.
  • The dominated convergence theorem gives sufficient conditions for almost sure convergence to imply L1-convergence:
\left. \begin{array}{ccc} X_n\xrightarrow{a.s.} X \\ \\ |X_n| < Y \\ \\ \mathrm{E}(Y) < \infty \end{array}\right\} \quad\Rightarrow \quad X_n\xrightarrow{L^1} X
  • A necessary and sufficient condition for L1 convergence is and the sequence (Xn) is uniformly integrable.

Read more about this topic:  Convergence Of Random Variables

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