What Is the Martingale Property in Wald's Equation?

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The discussion centers on the Martingale property in the context of Wald's equation, specifically questioning why the sequence Z_n = S_n - nμ is considered a martingale. The user expresses confusion regarding the proof that establishes this property for the sequence S_n, which is defined as the sum of independent and identically distributed (iid) random variables Y_i with a finite mean μ. Participants in the discussion seek clarification on the definition of a martingale being used to understand this proof better. The conversation highlights the need for a clear understanding of martingale definitions to resolve the confusion. Overall, the thread emphasizes the importance of foundational concepts in probability theory.
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Hi everyone. I was going through a proof of Wald's equation, where it was claimed that if {S_n} is a sequence defined as S_n = \sum_1^{n} Y_i where the Y_i are iid with finite mean \mu, then Z_n = S_n - n \mu is a martingale.

But I don't see why... at all!

Help!
 
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Epsilon36819 said:
Hi everyone. I was going through a proof of Wald's equation, where it was claimed that if {S_n} is a sequence defined as S_n = \sum_1^{n} Y_i where the Y_i are iid with finite mean \mu, then Z_n = S_n - n \mu is a martingale.

But I don't see why... at all!

Help!

What definition of Martingale are you using?
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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