Session Variance: Short-Term Volatility Dynamics

Session variance characterizes the dispersion of results within a limited series of iterations (session) and is a key parameter for assessing short-term risks. Unlike long-term variance, session variance depends significantly on the sample size n and can deviate significantly from the theoretical value. For a session of n iterations with a single iteration mathematical expectation μ and variance σ², the expectation of the total result is n·μ, and the variance of the sum is n·σ² (for independent iterations). The standard deviation of the total result σ_sum = σ·√n grows slower than the expected result n·μ, which ensures the stabilization of the relative deviation as n increases.

The influence of sample size on the reliability of the session variance estimate is described by the χ² (chi-squared) distribution. The sample variance s² = Σ(xᵢ − x̄)²/(n − 1) satisfies the relation (n − 1)·s²/σ² ~ χ²(n − 1). The confidence interval for the variance is: [(n − 1)·s²/χ²_{α/2,n−1}, (n − 1)·s²/χ²_{1−α/2,n−1}]. For small samples (n < 30), this interval is extremely wide, making the variance estimate unreliable. For example, at n = 20 and α = 0.05, the width of the 95% confidence interval for σ² is approximately [0.58σ², 2.11σ²], meaning the true variance can differ from the estimated one by 2–3 times.

To increase the reliability of the session variance estimate, robust statistics and resampling methods are applied. The bootstrap method allows for estimating the distribution of sample variance without assumptions about the parametric form of the initial distribution: B ≥ 1000 resamples with replacement are generated, s²_b is calculated for each, and the empirical distribution of the variance estimate is constructed. The percentile confidence interval [s²_{(α/2)}, s²_{(1−α/2)}] provides correct coverage even for heavy-tailed distributions. Alternatively, Winsorized variance and trimmed variance estimates, which are resistant to outliers, are used.

Practical analysis of session variance includes comparing the observed deviation with the theoretically expected one to identify anomalies. The Z-score of a session is calculated as Z = (S − n·μ) / (σ·√n), where S is the total result of the session. For normally distributed sums (with sufficient n by the CLT), values |Z| > 2.576 (α = 0.01) indicate statistically significant deviations. However, it is necessary to take into account the multiple comparisons effect: when analyzing m sessions, the probability of a false positive increases to 1 − (1 − α)ᵐ. The Bonferroni correction (α_adj = α/m) or the Benjamini-Hochberg (FDR) method allow controlling the false positive rate.