Wiener process continuous in probability

you can find the definition of a Wiener process at the the following link :

http://en.wikipedia.org/wiki/Wiener_process, in  the following discussion we will escape from the rigorous mathematical framework  in order to extract the main idea of the proof

it says that W(t) has independent increment : W(t) – W(s) ~ N(0 , t-s)  for 0 ≤ t < s where N is the famous normal distribution  : http://en.wikipedia.org/wiki/Normal_distribution

then we can rewrite for a small increment of t :  (*) W(t + Δt ) – W(t) = Φ √Δt,  where Φ is a random variable which follows N(0,1)

Therefore at time t , (meaning t is fixed,  the variable is Δt )  E[ W(t + Δt)  – W(t)] = 0

then E[ W( t + Δt)]  = W(t)

we can write for all ε > 0 Prob( W(t+ Δt) – W(t)  > ε) = Prob( W(t+ Δt) – E[ W( t + Δt)]  > ε),  we can use the Chebyshev inequality

Prob( W(t+ Δt) – E[ W( t + Δt)]  > ε) ≤ var[ W(t+ Δt) ]  / ε*ε

from (*) we got  var[ W(t+ Δt) ] = Δt

Finally  Prob( W(t+ Δt) – E[ W( t + Δt)]  > ε) ≤ Δt /  ε*ε

then  for all   ε > 0  Prob( W(t+ Δt) – E[ W( t + Δt)]  > ε) = 0 when Δt → 0

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