Αποτελέσματα Αναζήτησης
Proposition 1. Let {Xn} be a sequence of random variables, not necessarily independent. (i) If P∞ E[|X a.s. n| s] < ∞, and s > 0, then Xn → 0. n=1 (ii) If. P. ∞ a.s. n=1. P(|X. n | > ǫ) < ∞, for every ǫ > 0, then X. n. → 0. (iii) X. a.s. n m. → 0 iff for every ǫ > 0 we have P[sup. ≥n |X. n | > ǫ] → 0 as n → ∞. Proof.
Let X and Xn, n ∈ N, be random variables with CDFs F and Fn, respectively. We say that the sequence Xn converges to X in distribu tion, and write Xn d → X, if. for every x ∈ R at which F is continuous. (a) Recall that CDFs have discontinuities (“jumps”) only at the points that have positive probability mass.
n converges to X in probability, written X n!p X, if, for every †>0, P(jX n ¡Xj >†)! 0 as n !1. Let F n denote the cdf of X n and let F denote the cdf of X. X n converges to X in distribution, written X n!d X, if, lim n F n(t)=F(t) at all t for which F is continuous. Here is a summary: Quadratic Mean E(X n ¡X)2! 0 In probability P(jX n ...
Let X and Xn, n ∈ N, be random variables with CDFs F and Fn, respectively. We say that the sequence Xn converges to X in distribu-tion, and write Xn d → X, if. for every x ∈ R at which F is continuous. (a) Recall that CDFs have discontinuities (“jumps”) only at the points that have positive probability mass.
For a random variable X, the event {! | X(!) ≤ c} is often written as {X ≤ c}, and is sometimes just called “the event that X c.” The probability of this event is well defined, since this event belongs to ≤. F.
Suppose |Xn| ≤ Y with E(Y ) < ∞, and Xn → X in probability. Show that E|Xn − X| → 0 (i.e., Xn → X in L1). Solution. Method 1. Suppose Xn 9 X in L1. Then, there exists a subsequence {nk} such that E|Xnk − X| > ǫ for some ǫ > 0 and all {nk}. Since Xnk → X in probability, there exists a further subsequence nkj such that Xnkj → X a.e..
P(X 1=n); thus P(X 1=n) = 0, so P(X>0) = P 0 @ \ n 1 fX 1=ng 1 A= lim n!1 P(X 1=n) = 0: (d) follows immediately from (a). The following lemma gives a way to approximate nonnegative random variables with monotone sequences of simple ones. 1.3 Lemma. If Xis a nonnegative random variable, then there is a sequence (Z n) of
