> ## Documentation Index
> Fetch the complete documentation index at: https://nixtlaverse.nixtla.io/llms.txt
> Use this file to discover all available pages before exploring further.

# Levy process (heavy tails)

A Levy (alpha-stable) process generalizes Brownian motion to allow
*heavy tails* and jumps: increments are independent but drawn from a
stable distribution whose tail weight is tunable. It models series with
occasional extreme moves that a Gaussian model badly underestimates.

> **The model**
>
> $y_t = y_{t-1} + \text{scale}\cdot X_t + \text{location}, \qquad X_t \sim S(\alpha, \beta; 1)$
>
> The stability index `alpha` ∈ (0, 2] sets the tail heaviness:
> `alpha = 2` recovers Gaussian increments, while smaller values give
> progressively heavier tails and larger jumps (`alpha = 1` is
> Cauchy-like, with undefined variance).

```python theme={null}
import polars as pl
import matplotlib.pyplot as plt

from synforecast.generators import LevyProcessGenerator
```

## 1. Tail heaviness

Lower alpha = heavier tails = more extreme jumps.

```python theme={null}
fig, axes = plt.subplots(3, 1, figsize=(12, 9), sharex=True)

for ax, alpha, label in zip(
    axes,
    [2.0, 1.5, 1.0],
    ["alpha=2.0 (Gaussian)", "alpha=1.5 (heavy-tailed)", "alpha=1.0 (Cauchy-like)"],
):
    gen = LevyProcessGenerator(engine="polars", 
        min_length=500, max_length=500, freq="D",
        alpha=alpha, cumulative=False, seed=42,
    )
    df = gen.generate(n_series=1)
    ax.plot(df["ds"].to_list(), df["y"].to_list(), alpha=0.8, linewidth=0.8)
    ax.set_ylabel("Increment")
    ax.set_title(label)

axes[-1].set_xlabel("Timestamp")
plt.tight_layout()
plt.show()
```

<img src="https://mintcdn.com/nixtla/kY7DsWHSCHfeJfFk/synforecast/docs/generators/stochastic/levy_process_files/figure-markdown_strict/cell-3-output-1.png?fit=max&auto=format&n=kY7DsWHSCHfeJfFk&q=85&s=ab3995f9475536e1f237456c58be4773" alt="" width="1189" height="889" data-path="synforecast/docs/generators/stochastic/levy_process_files/figure-markdown_strict/cell-3-output-1.png" />

## 2. Multiple series

```python theme={null}
multi_gen = LevyProcessGenerator(engine="polars", 
    min_length=200, max_length=200, freq="D",
    alpha=1.8, cumulative=True, seed=42,
)
multi_df = multi_gen.generate(n_series=3)

fig, ax = plt.subplots(figsize=(12, 4))
for uid in multi_df["unique_id"].unique().to_list():
    series = multi_df.filter(pl.col("unique_id") == uid)
    ax.plot(series["ds"].to_list(), series["y"].to_list(), label=uid, alpha=0.8)
ax.set_xlabel("Timestamp")
ax.set_ylabel("Value")
ax.set_title("Multiple Levy process series (alpha=1.8)")
ax.legend()
plt.tight_layout()
plt.show()
```

<img src="https://mintcdn.com/nixtla/kY7DsWHSCHfeJfFk/synforecast/docs/generators/stochastic/levy_process_files/figure-markdown_strict/cell-4-output-1.png?fit=max&auto=format&n=kY7DsWHSCHfeJfFk&q=85&s=480c53f2e73144a56ef45973d69d16a4" alt="" width="1190" height="390" data-path="synforecast/docs/generators/stochastic/levy_process_files/figure-markdown_strict/cell-4-output-1.png" />

> **Related generators**
>
> * [Jump diffusion](jump_diffusion) — a smooth diffusion with
>   separate discrete jumps.
> * [GARCH](garch) — heavy-tailed *conditional* behavior via
>   volatility clustering.
>
> The stability parameter is documented in the [generator
> reference](https://github.com/Nixtla/synforecast/blob/main/GENERATORS.md).