To address the question, we first clarify potential ambiguities. The original definition of winnum as log(tstarstar) - log(tstar) is problematic because tstarstar > tstar implies winnum is always positive, contradicting the claim that it could be negative. We assume this is a typo and winnum should be log(x_{tstarstar}) - log(x_{tstar}), which depends on the ratio of the ending and ini...

Short answer: Your statements A and B are false. You can construct sequences where the crossing happens in fewer than n steps but the log-return is negative, and also sequences where the crossing takes at least n steps but the log-return is positive. So there’s no such deterministic relation in general.

First, a quick clarification: you wrote winnum = log(tstarstar) − log(tstar). If tstarstar > t...

This is an interesting problem involving moving averages and logarithms. Let me analyze your claims carefully.

First, let me clarify the setup:

• You have a sequence of positive numbers
• At time , you calculate
• The game starts when
• The game ends when $\log(x_{t^{*...

The statements you call A) and B) are unfortunately not true in general.
Below are two explicit counter-examples that violate them, followed by an explanation of why no “time–length versus profit” rule of the type you proposed can hold.

Notation that will be used:

Lk = log(xk) (the samples that are actually averaged)
Mk = (1/n) Σi=0…n-1 Lk-i (the n-point moving average of the lo...

The statement is unfortunately not correct in general; it becomes false as soon as the length of the moving–average window is larger than 2.
Below is an explicit counter-example for the case n = 3 (the same idea works for any n ≥ 3). After the example you will find an explanation of why the claim fails and a remark on the only case in which it is true (n = 2).

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The game has two parameters: n (window size) and d (a positive threshold).

The player starts the game at time t* (tstar) when: log(...

This is a fascinating question that sits right at the intersection of time-series analysis, signal processing, and even quantitative finance. The relationship you're exploring is a simplified model of a "mean reversion" strategy.

Let's break down your claim and see if we can validate it or find a counter-example.

Formalizing the Problem

First, let's establish a clear mathematical notation t...

Here is a very simple “by‐hand” counter–example which shows that neither of the two implications

(i) if t**–t* < n then winnum>0, and t**–t* ≥ n then winnum≤0
(ii) if winnum>0 then t**–t* < n, and if winnum≤0 then t**–t* ≥ n

holds in general. In fact one can do everything in log‐space so that xᵢ>0 is automatic.

— Take n=2. Let L₁=L₂=0 so that the 2-point moving average at t=2 ...

Moving Average Relation Analysis

I'll analyze the claimed relationship between game duration and player's winnings in this moving average problem.

Understanding the Setup

Let me first establish the key elements:

• Positive numbers are generated sequentially
• At time , we calculate (moving average with window size of log values)
• Game...