Random Evolutionary Systems

8Diffusion Approximation of Random Evolutions in Random Media

Discrete–continuous time is determined by the connection of discrete instants of time k ≥ 0, k ∈ ℕ := {0, 1, ...} with continuous time t ≥ 0, t ∈ ℝ₊ := [0, +∞) by the formula

[8.1]

The integer part of the product means that

[8.2]

The transition from the discrete instants of time k ≥ 0 to the continuous time t ≥ 0 means that when increasing the sample size N → ∞, the intervals between adjacent moments of continuous time tend to 0:

[8.3]

However, the condition

[8.4]

also means that k → ∞.

8.1. Binary discrete Markov evolutions

As previously discussed, statistical experiments are defined by the averaged sums of sample variables (δ_n(k), 1 ≤ n ≤ N ), k ≥ 0, jointly independent for fixed k ≥ 0 and identically distributed for different n ∈ [1, N ], taking two values ±1

[8.5]

The binary statistical experiment [8.5] can also specify the difference between the frequency statistical experiments (see section 1.1):

[8.6]

In the general ...