Problem regarding the reversed martingale

Reference no: EM13966749

Let Y be a random variable independent of all the X j , with E |Y | <>∞. Let Uj := Y + Tj for j = -1, -2,... . Let Ck be the smallest σ-algebra for which Uj is measurable for all j ≤ k. Show that {Uk, Ck }k≤-1 is a reversed martingale and find its limit as k → -∞. Let C-∞ := nk≤-1 Ck . Is Y equal a.s. to a variable measurable for C-∞?

Reference no: EM13966749

Questions Cloud

Second-countable topological space : Let (S, T ) be a second-countable topological space and (Y, d) any metric space. Show that the Borel σ-algebra in the product S × Y is the product σ-algebra of the Borel σ-algebras in S and in X.

Determining the complements of sets : Let R be a σ-ring of subsets of a set X . Let S be the σ-algebra generated by R. Recall (§3.3, Problem 8) or prove that S consists of all sets in R and all complements of sets in R.

Provide directions for your examples : What step would you add? Provide 2 examples for your classmates to attempt. Provide directions for your examples.

Find the acceleration at seconds : Suppose that an automobile's velocity starting from rest is v(t) = 270t/3t + 11 where v is measured in feet per second. Find the acceleration at 7 seconds. Round your answer to one decimal place.

Problem regarding the reversed martingale : Let Ck be the smallest σ-algebra for which Uj is measurable for all j ≤ k. Show that {Uk, Ck }k≤-1 is a reversed martingale and find its limit as k → -∞. Let C-∞ := nk≤-1 Ck . Is Y equal a.s. to a variable measurable for C-∞?

What is the total line current il when the switch is open : The figure shown below shows a one-line diagram of a simple power system containing a single 480 V generator and three loads. Assume that the transmission lines in this power system are lossless, and answer the following questions. Assume that..

What are the potential obstacles of your team : How would you ensure each individual felt like a vital member of the team? Summarize the steps you would take to build trust in leading this team.

Three-series theorem : (a) Give an example where Sn converge a.s., but are not L1-bounded. (b) If |X j |≤ 2 a.s. for all j , and Sn converges a.s., show that Sn converges in L2. Hint: Use the three-series theorem (9.7.3).

Compute the acceleration at time t : Compute the length of its path from t = 0 to t = 5, Compute the velocity at time t and compute the acceleration at time t.

User Account

All Pages