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The Wiener process is defined by the three properties: 1. W(0) $W(0) = 00$, 2. W(t) $W(t)$ is almost surely continuous, and 3. W(t) $W(t)$ has independent increments with W(t) $W(t) - W(s) ~ \sim N(0, t-s) t-s)$ (for $0 ≤ s < t)t$).

What would be an example of a process which satisfies 1) and 3), but not 2) ?

I am going to teach an introductory class on Brownian motion at advanced undergrad level. Just wanted to make sure that all the conditions are mutually independent.

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# WeinerWiener process related counterexample

The Weiner Wiener process is defined by the three properties: 1. W(0) = 0, 2. W(t) is almost surely continuous, and 3. W(t) has independent increments with W(t) - W(s) ~ N(0, t-s) (for 0 ≤ s < t).

What would be an example of a process which satisfies 1) and 3), but not 2) ?

I am going to teach an introductory class on Brownian motion at advanced undergrad level. Just wanted to make sure that all the conditions are mutually independent.

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# Weiner process related counterexample

The Weiner process is defined by the three properties: 1. W(0) = 0, 2. W(t) is almost surely continuous, and 3. W(t) has independent increments with W(t) - W(s) ~ N(0, t-s) (for 0 ≤ s < t).

What would be an example of a process which satisfies 1) and 3), but not 2) ?

I am going to teach an introductory class on Brownian motion at advanced undergrad level. Just wanted to make sure that all the conditions are mutually independent.