Stochastic simulations are used to create synthetic one-dimensional telegraph approximation (TA) signals based on turbulent zero crossings, where the interval between crossings is governed by a power-law probability distribution with exponent $\alpha$. The power-law exponent is determined for statistics of simulated TA signals, namely, the box-counting fractal dimension $D_1$, the energy spectrum exponent ${\beta }_{\mathrm{TA}}$, and the intermittency exponent ${\mu }_{\mathrm{TA}}$. For the binary TA signal with no variability in amplitude, the parameters are related linearly as $D_1=2-{\beta }_{\mathrm{TA}}=1-{\mu }_{\mathrm{TA}}$. The relations are unchanged if the crossing interval distribution has a finite power-law region (i.e., inertial subrange) representing a flow with a finite Reynolds number. However, the finite distribution yields statistics that are not truly scale invariant and distorts the linear relation between the statistic exponents and $\alpha$. The behavior is due to finite-size effects apparent from the survival function, or the complementary cumulative distribution, which for finite Reynolds number is only approximately self-similar and has an effective exponent differing from $\alpha$. An expression presented for the effective exponent recovers the expected relations between $\alpha$ and the TA statistics. The findings demonstrate how a finite Reynolds number can affect indicators of self-similarity, fractality, and intermittency observed from single-point measurements.

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• Self-organized systems
• Turbulence
• Fractal dimension characterization
• Stochastic analysis
• Time series analysis