What is the Role of Epsilon in Stochastic Continuity?

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SUMMARY

The discussion clarifies the role of epsilon (ε) in the context of stochastic continuity, emphasizing that ε is a small positive number used as an upper bound for the difference between two random variables. Participants highlight the distinction between intuitive visualizations and formal mathematical definitions, asserting that ε does not reside on any axis but rather represents a numerical value in the analysis of continuity. The conversation also touches on the concept of jump discontinuity in right-continuous functions, specifically referencing RCLL functions.

PREREQUISITES
  • Understanding of stochastic processes and their properties
  • Familiarity with the concept of limits in calculus
  • Knowledge of right-continuous functions and jump discontinuities
  • Basic grasp of random variables and their significance in probability theory
NEXT STEPS
  • Study the formal definition of stochastic continuity in detail
  • Explore the implications of jump discontinuities in stochastic processes
  • Learn about RCLL (right-continuous with left limits) functions and their applications
  • Investigate the role of epsilon in limit definitions within calculus
USEFUL FOR

Mathematicians, statisticians, and students of probability theory who are looking to deepen their understanding of stochastic continuity and the role of epsilon in mathematical definitions.

woundedtiger4
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ε is an arbitrary (small) number > 0.
If you are hung up on using axes, s and t are points on the x axis. Xt is a point on the y axis, but it is a random variable rather than just a number.
 
Then, is epsilon on y-axis?
 
woundedtiger4 said:
Then, is epsilon on y-axis?

You are confusing intuitive visualizations of mathematics with the content of mathematical definitions. Even in calculus, there is nothing in the definition of limit that says that epsilon in "on the y-axis".
 
I know what you mean actually I can't understand without visualising therefore it is irritating me that what is epsilon intuitively in continuity of stochastic process? I know the op is about jump discontinuity which is RCLL function so is the epsilon shows any point between the jump ?
 
woundedtiger4 said:
Then, is epsilon on y-axis?

ε is a positive number. It is used as an upper bound of the magnitude difference of two random variables. There is no axis involved. If you insist on thinking "axis", then you may consider everything on the y axis. However, I suggest you try to understand the main point, there is no axis involved, just numbers.
 
  • #10
No, I wasn't thinking epsilon on y-axis,I just tried to give an example that like in calculus I used to think epsilon (not the same epsilon shown in stochastic continuity) as on y-axis for which the delta exists. I didn't mean that the epsilon in stochastic continuity is on y-axis .
Thanks a tonne because now I have understood it.
 

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