# Random variables and probability distributions, Edition: by H. Cramer

By H. Cramer

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Extra info for Random variables and probability distributions, Edition: 3ed.

Example text

For r > 0, define/(co) = X{j}(T((y))*/(a;). )(co)=/(^)(co,(J for all ca e Q. 13) /(co)=/(co,(J T(CO^(^)). 12) for all co e Q. 11) holds. ) for o) e Q, which explicitly displays/as a o'[x(s AT): S > 0] measurable function. D 34 1. 14) eco'M) = Z > ' ) for all AGM,. 2. Then it is clear that Q^ is again a conditional probability distribution of P given Jt^.. 14) holds for all co'. 4 Theorem. 14) holds for alio)'. d. of P\M^. 15) QioK^{^) — ^(<^') and x(s, co) = x(s, a>') for 0 < s < T(CO)) = 1. 5 Theorem.

Proof. , [T^(„)_I(CO), T^(e«)(co)), aud [XN(CO)(O^\ T]. All of these subintervals, except possibly the last one, must have length greater than S^(p). Thus, either both ti and t2 lie in the same subinterval, or they He in adjacent subintervals. 4. Martingales and Compactness 37 equal to p/2. Hence the oscillation over the union of two successive subintervals cannot exceed p. In particular, \x(t2, co) - x(fi, a))\ < p. 2) P({co: SM < S}). 2 Hypothesis. For all non-negative f e C^lR'^) there is a constant Af >0 such that (f(x(t)) + Aft, Ml, P) is a non-negative submartingale.

8. Suppose (9(t), i^,, P) is a martingale on (£, i^, P). ^, P). ) Show that {6(t), ^t+o, P) is a. martingale. 9. 8 to show that if (9(t), #",, P) is a continuous real valued martingale which is almost surely of bounded variation, then for almost all q, 6(t) is a constant in t. Note that this conclusion is definitely false if one drops the assumption of continuity. 10. Suppose (9{t), ^,, P) is a martingale on (£, J^, P) such that sup,^o £(^(0)^ ^ ^' Show that E{9{t))^ is an increasing function of ?