Econophysics and Interdisciplinary Applications

Econophysics

Econophysics is an interdisciplinary field of research that applies the theories, concepts, and computational methods of physics — especially those of statistical mechanics, complex systems, and nonlinear dynamics — to the study of economic and financial phenomena. The term was coined by the American physicist H. Eugene Stanley in 1996 (at the Kolkata conference on "Econophysics"), though the approach itself dates back to work such as that of Benoit Mandelbrot on price fluctuations in the 1960s.

The central idea is that an economy is a complex system consisting of a very large number of interacting agents (firms, traders, consumers), much like a gas of interacting molecules. Therefore, the same mathematical tools used to describe many-body physical systems can be used to describe economic systems.

Motivation of Physicists toward Economics

Physicists are motivated to enter economics for several reasons:

1. Similarity of the systems. Both physical and economic systems are many-body systems with a huge number of interacting units. Concepts such as random walks, diffusion, scaling, power laws, phase transitions, and self-organized criticality appear naturally in both.

2. Availability of large data. Modern financial markets generate enormous amounts of high-frequency data (tick-by-tick prices, order books). This "data-rich" environment is attractive to physicists trained in data analysis and statistical mechanics.

3. Failure of standard economic models. Traditional economic models assume rational agents and Gaussian (normal) distributions. Empirical studies show that financial returns have fat tails, volatility clustering, and long-range correlations — features well known in physics but poorly captured by classical economics. Physicists are motivated to build more realistic models.

4. Universal laws. Physicists look for universal statistical laws. For example, the distribution of price changes, the distribution of firm sizes, and the distribution of income follow power laws similar to those in physical systems.

5. Methodological transfer. Tools such as Monte Carlo simulation, percolation theory, spin models (Ising model), agent-based modeling, and network theory can be directly applied to economic problems.

6. Interdisciplinary appeal. Econophysics lies at the boundary of physics, mathematics, economics, and computer science. It offers a rich research area where physical intuition can solve real economic problems, making it an attractive interdisciplinary research topic.

Importance: Econophysics provides a new perspective on market dynamics, risk management, and financial crashes (e.g., the 2008 crisis), and it has produced useful models such as the Minority Game and Cont-Bouchaud model of market fluctuations.

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OR

Berry-Esseen Theorem

The Berry-Esseen theorem (also called the Berry-Esseen inequality) is a refinement of the Central Limit Theorem (CLT). While the CLT tells us that the sum of independent and identically distributed (i.i.d.) random variables converges in distribution to a normal distribution, it does not tell us how fast this convergence occurs. The Berry-Esseen theorem provides a quantitative bound on the rate of convergence.

Statement I (Classical Form)

Let $X_1, X_2, \dots, X_n$ be independent and identically distributed random variables with

$E[X_i] = \mu, \qquad \text{Var}(X_i) = \sigma^2 > 0, \qquad E\big[|X_i - \mu|^3\big] = \rho < \infty.$

Define the standardized sum

$Z_n = \frac{\sum_{i=1}^{n} X_i - n\mu}{\sigma\sqrt{n}}.$

Let $F_n(x)$ be the cumulative distribution function (CDF) of $Z_n$, and let $\Phi(x)$ be the CDF of the standard normal distribution. Then there exists a universal constant $C$ (independent of $n$ and of the distribution of $X_i$) such that for all $x$ and all $n$,

$\sup_{x \in \mathbb{R}} \big| F_n(x) - \Phi(x) \big| \le \frac{C\,\rho}{\sigma^3 \sqrt{n}}.$

Statement II (Sharpened / Modern Form)

The constant $C$ has been improved over time. The best known bound (Shevtsova, 2011) is

$C \le 0.5600,$

so that

$\sup_{x} \big| F_n(x) - \Phi(x) \big| \le \frac{0.5600\,\rho}{\sigma^3 \sqrt{n}}.$

Equivalently, in terms of the third absolute moment $\beta_3 = E[|X_i - \mu|^3]$,

$\sup_{x} \big| F_n(x) - \Phi(x) \big| \le \frac{C\,\beta_3}{\sigma^3 \sqrt{n}}.$

The theorem also holds in a non-identically distributed form (Lyapunov-type version), where the bound involves the sum of third absolute moments.

Explanation of the Terms

• $F_n(x) - \Phi(x)$: the error between the true distribution of the standardized sum and the normal distribution.
• $\sup_x$: the supremum (maximum) of this error over all real $x$ — the Kolmogorov distance.
• $\rho / \sigma^3$: a dimensionless measure of skewness of the underlying distribution.
• $1/\sqrt{n}$: the rate of convergence — the error decreases as the sample size increases.

Importance of the Berry-Esseen Theorem

1. Rate of convergence. It tells us that the error in the normal approximation decreases at the rate $O(1/\sqrt{n})$, which the CLT alone does not provide.

2. Validity of normal approximation. It gives a practical criterion for when the normal approximation is reliable: the approximation is good when $n$ is large and the skewness $\rho/\sigma^3$ is small.

3. Confidence intervals and hypothesis testing. In statistics, it justifies the use of normal-based confidence intervals and $t$-tests for sample means, by bounding the error introduced when the population is not normal.

4. Non-asymptotic results. Unlike the CLT (which is asymptotic), the Berry-Esseen bound holds for finite $n$, making it useful in small-sample problems.

5. Foundation for further results. It is a fundamental tool in probability theory and has led to generalizations such as the Berry-Esseen theorem for martingales, for dependent variables, and in higher dimensions.

6. Applications. It is used in actuarial science, finance (risk estimation), physics (statistical mechanics), and computer science (randomized algorithms) where sums of many random variables are approximated by normal distributions.

Central Limit Theorem (CLT)

Statement

Let $X_1, X_2, \dots, X_n$ be a sequence of independent and identically distributed (i.i.d.) random variables, each with finite mean $\mu$ and finite variance $\sigma^2$. Then, as the sample size $n$ becomes large, the sampling distribution of the sample mean $\bar{X}$ (or equivalently the sum $S_n = \sum_{i=1}^{n} X_i$) approaches a normal distribution, regardless of the shape of the population distribution.

Derivation

Let the population have mean $\mu$ and variance $\sigma^2$. Define the sample mean:

$\bar{X} = \frac{X_1 + X_2 + \dots + X_n}{n}$

Step 1: Mean of the sample mean

$E(\bar{X}) = E\left(\frac{1}{n}\sum_{i=1}^{n} X_i\right) = \frac{1}{n}\sum_{i=1}^{n} E(X_i) = \frac{1}{n}(n\mu) = \mu$

Step 2: Variance of the sample mean

Since the $X_i$ are independent,

$\text{Var}(\bar{X}) = \text{Var}\left(\frac{1}{n}\sum_{i=1}^{n} X_i\right) = \frac{1}{n^2}\sum_{i=1}^{n} \text{Var}(X_i) = \frac{1}{n^2}(n\sigma^2) = \frac{\sigma^2}{n}$

So the standard error of the mean is $\sigma_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}}$.

Step 3: Standardization

Define the standardized variable:

$Z_n = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} = \frac{S_n - n\mu}{\sigma\sqrt{n}}$

Step 4: Moment generating function approach

Let $Y_i = \dfrac{X_i - \mu}{\sigma}$, so that $E(Y_i) = 0$ and $\text{Var}(Y_i) = 1$. Then

$Z_n = \frac{1}{\sqrt{n}}\sum_{i=1}^{n} Y_i$

The moment generating function (MGF) of $Z_n$ is:

$M_{Z_n}(t) = E\left[e^{tZ_n}\right] = E\left[\exp\left(\frac{t}{\sqrt{n}}\sum_{i=1}^{n} Y_i\right)\right] = \prod_{i=1}^{n} E\left[e^{tY_i/\sqrt{n}}\right] = \left[M_Y\left(\frac{t}{\sqrt{n}}\right)\right]^n$

Step 5: Taylor expansion of the MGF

Expanding $M_Y(u)$ about $u = 0$:

$M_Y(u) = 1 + u\,E(Y) + \frac{u^2}{2}E(Y^2) + o(u^2) = 1 + \frac{u^2}{2} + o(u^2)$

since $E(Y) = 0$ and $E(Y^2) = 1$. Putting $u = t/\sqrt{n}$:

$M_Y\left(\frac{t}{\sqrt{n}}\right) = 1 + \frac{t^2}{2n} + o\left(\frac{1}{n}\right)$

Step 6: Taking the limit

$M_{Z_n}(t) = \left[1 + \frac{t^2}{2n} + o\left(\frac{1}{n}\right)\right]^n \xrightarrow{n \to \infty} e^{t^2/2}$

This is exactly the MGF of the standard normal distribution $N(0,1)$. By the uniqueness theorem of MGFs,

$Z_n \xrightarrow{d} N(0,1)$

Hence, for large $n$,

$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{approximately}$

Essential Conditions

1. Independence: The observations $X_1, X_2, \dots, X_n$ must be independent of one another.
2. Identical distribution: All observations must come from the same population (same distribution).
3. Finite mean and variance: The population must have a finite mean $\mu$ and finite variance $\sigma^2$ (i.e., $E(X^2) < \infty$).
4. Large sample size: The sample size $n$ must be sufficiently large (usually $n \ge 30$ is taken as sufficient).
5. Random sampling: The sample must be drawn randomly so that each observation is representative of the population.

Significance in Finance

1. Portfolio Return Modelling: The return on a diversified portfolio is essentially a weighted average of many individual asset returns. By CLT, portfolio returns tend to be approximately normally distributed, which justifies using mean–variance analysis (Markowitz) and Value at Risk (VaR) models.

2. Risk Measurement (VaR and Expected Shortfall): VaR is computed from the normal distribution of returns. CLT provides the theoretical basis for assuming normality of aggregated returns, allowing risk managers to estimate the maximum expected loss at a given confidence level.

3. Option Pricing (Black–Scholes Model): The Black–Scholes model assumes that log-returns of stock prices are normally distributed. Since log-returns over many small time intervals are sums of i.i.d. increments, CLT supports this normality assumption.

4. Sampling and Estimation: Financial analysts use sample means (e.g., average return, average default rate) to estimate population parameters. CLT gives the sampling distribution of these estimates, enabling construction of confidence intervals and hypothesis tests.

5. Insurance and Actuarial Pricing: Insurers pool a large number of independent risks. CLT ensures that the average claim per policyholder is approximately normal, so premiums and reserves can be set using normal-based probability calculations.

6. Diversification Benefit: CLT explains why diversification reduces risk: as $n$ increases, $\text{Var}(\bar{X}) = \sigma^2/n \to 0$, so portfolio risk falls with more assets.

Thus CLT is a cornerstone of modern financial theory, risk management, and quantitative finance.

Time Series Data — Definition and Examples

Definition: Time series data is a sequence of observations of a variable recorded at successive points in time, usually at equal intervals. In symbols, a time series is $\{X_t\}$, where $t = 1, 2, 3, \dots, n$ denotes time and $X_t$ is the value of the variable at time $t$.

Examples:

1. Economic: Nepal's annual GDP from 2000 to 2024; monthly inflation rates of Nepal Rastra Bank.
2. Financial: Daily closing price of NEPSE index; weekly exchange rate of Rs per US dollar.
3. Business: Yearly sales of a company; quarterly profit figures.
4. Physical/Social: Daily temperature of Kathmandu; yearly population of Nepal.

Information contained in a financial time series:

1. Trend (long-term movement): The general upward or downward tendency of prices, e.g., a steadily rising NEPSE index over years.
2. Seasonal variation: Regular fluctuations within a year, e.g., higher share trading near the fiscal year-end.
3. Cyclical variation: Long-wave movements of more than one year, e.g., boom and recession phases in the stock market.
4. Irregular (random) variation: Unpredictable shocks such as political instability, earthquakes, or policy changes.
5. Volatility: The degree of fluctuation in returns, which measures financial risk.
6. Serial dependence (autocorrelation): The dependence of present values on past values, which allows forecasting.
7. Stationarity/non-stationarity: Whether the mean, variance, and covariance remain constant over time — crucial for model selection (ARIMA, ARCH, etc.).

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OR

Show that for a stationary stochastic process: $C(\tau) \sim R(\tau)$

Let $\{X_t\}$ be a stationary stochastic process with constant mean $\mu$ and constant variance $\sigma^2$.

Definitions:

Autocovariance at lag $\tau$: $C(\tau) = E\left[(X_t - \mu)(X_{t+\tau} - \mu)\right]$

Autocorrelation at lag $\tau$: $R(\tau) = \frac{E\left[(X_t - \mu)(X_{t+\tau} - \mu)\right]}{\sqrt{E(X_t-\mu)^2}\sqrt{E(X_{t+\tau}-\mu)^2}}$

Proof:

Since the process is stationary, the variance is constant for all $t$: $E(X_t - \mu)^2 = E(X_{t+\tau} - \mu)^2 = \sigma^2$

Substituting in the definition of $R(\tau)$: $R(\tau) = \frac{E\left[(X_t - \mu)(X_{t+\tau} - \mu)\right]}{\sqrt{\sigma^2}\sqrt{\sigma^2}} = \frac{C(\tau)}{\sigma^2}$

Hence, $C(\tau) = \sigma^2 R(\tau) \quad \text{and} \quad R(\tau) = \frac{C(\tau)}{C(0)}$

since $C(0) = E(X_t-\mu)^2 = \sigma^2$.

Conclusion: The autocovariance $C(\tau)$ and autocorrelation $R(\tau)$ differ only by the constant factor $\sigma^2$ (the variance of the process). Therefore they are proportional to each other, written as $C(\tau) \sim R(\tau)$. Both have the same shape and the same correlogram pattern; $R(\tau)$ is simply the standardized (unit-free) form of $C(\tau)$, lying between $-1$ and $+1$.

Attempt any two of the following:

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(a) Brief note on the Chaos approach in market

The chaos approach in market analysis is based on the idea that financial markets are nonlinear, dynamic and deterministic systems whose long-run behaviour is highly sensitive to small changes in initial conditions. It is an offshoot of chaos theory, popularised in economics by studies such as those of Benoit Mandelbrot on fractal price movements.

Key features:

1. Nonlinearity: Market prices do not move in simple straight-line or proportional ways; they respond nonlinearly to news, expectations and feedback.
2. Sensitive dependence on initial conditions (the "butterfly effect"): A tiny change in an initial price, policy or expectation can produce large and unpredictable future price movements. Hence long-term exact forecasting is impossible.
3. Deterministic but unpredictable: The system follows definite rules (e.g., demand–supply, arbitrage, speculation), yet the resulting path looks random. This is why markets appear "random" even though they are not truly random.
4. Strange attractors: Price series tend to be pulled towards certain regions or patterns (attractors) without ever repeating exactly, producing fractal or self-similar patterns at different time scales.
5. Bifurcation and sudden shifts: A gradual change in a parameter (e.g., interest rate) can suddenly change the market's qualitative behaviour — from stable to volatile, or from a bull to a bear phase.
6. Implications for finance: Traditional efficient-market and linear econometric models fail to capture such behaviour. Chaos theory suggests that markets are partly predictable in the short run but not in the long run, so risk management, portfolio diversification and nonlinear models (e.g., ARCH, neural networks) become important.

Conclusion: The chaos approach views the market as a complex, feedback-driven, nonlinear system in which order and disorder coexist. It explains volatility clustering, bubbles and crashes better than purely random-walk models.

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(b) Definition of Martingales with example

Definition: A stochastic process $\{X_n, n \ge 0\}$ is called a martingale with respect to a filtration $\{\mathcal{F}_n\}$ if:

1. $E[|X_n|] < \infty$ for all $n$ (integrability),
2. $X_n$ is $\mathcal{F}_n$-measurable (adaptedness),
3. $E[X_{n+1} \mid \mathcal{F}_n] = X_n$ for all $n$ (fair-game property).

In words, a martingale is a process whose expected future value, given all present and past information, equals its present value. It models a "fair game" — no systematic gain or loss is expected.

Variants:

• Submartingale: $E[X_{n+1} \mid \mathcal{F}_n] \ge X_n$ (favourable game).
• Supermartingale: $E[X_{n+1} \mid \mathcal{F}_n] \le X_n$ (unfavourable game).

Example (simple random walk): Let $Y_1, Y_2, \dots$ be independent random variables with $P(Y_i = +1) = P(Y_i = -1) = \tfrac{1}{2}.$ Define $X_0 = 0$ and $X_n = Y_1 + Y_2 + \cdots + Y_n$.

Then $E[X_{n+1} \mid X_0, X_1, \dots, X_n] = E[X_n + Y_{n+1} \mid X_n] = X_n + E[Y_{n+1}] = X_n + 0 = X_n.$

Hence $\{X_n\}$ is a martingale. It represents the fortune of a gambler in a fair coin-tossing game: on average, the fortune neither grows nor shrinks.

Another example: If $S_n$ is the price of a stock in a fair market and $S_n$ is integrable, then $\{S_n\}$ is a martingale — the best prediction of tomorrow's price is today's price.

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(c) Markov and Non-Markov processes

Markov process: A stochastic process $\{X_n\}$ is called a Markov process (or Markov chain, in the discrete case) if the future depends only on the present, not on the past. Formally, for all states $i, j$ and times $n$, $P(X_{n+1} = j \mid X_n = i, X_{n-1}, \dots, X_0) = P(X_{n+1} = j \mid X_n = i).$ This is the Markov property or memorylessness.

Example: A random walk on integers, or the sequence of weather states (sunny/rainy) where tomorrow's weather depends only on today's.

Non-Markov process: A process is non-Markov if the future depends on more than just the present state — i.e., on the past history as well. Formally, $P(X_{n+1} = j \mid X_n = i, X_{n-1}, \dots, X_0) \ne P(X_{n+1} = j \mid X_n = i).$

Example: A process where $X_{n+1} = X_n + X_{n-1} + \varepsilon_n$ (an AR(2) type process). Here the next value depends on both $X_n$ and $X_{n-1}$, so the past matters — it is non-Markov.

Comparison:

Note: A non-Markov process can sometimes be converted into a Markov process by enlarging the state space (e.g., taking the pair $(X_n, X_{n-1})$ as the new state).

Arbitrage

Arbitrage is the practice of simultaneously buying and selling the same asset (or equivalent assets) in two different markets to profit from a price difference. The trader buys in the cheaper market and sells in the dearer market at the same time, so the position is riskless — the profit is locked in the moment both trades are executed. For arbitrage to exist, three conditions must hold:

1. The same asset (or a perfect substitute) must trade in two markets.
2. The two markets must be separated in time or space (so prices can differ).
3. The price difference must exceed the transaction cost.

Illustration with an example

Suppose the exchange rate between the Nepali rupee and the US dollar is quoted differently in two markets:

• Kathmandu market: Rs 130 = 1 USD
• New York market: Rs 132 = 1 USD

An arbitrageur with Rs 1,30,00,000 would:

Because both trades are done at the same instant at known prices, the arbitrageur bears no exchange-rate risk. The only costs are brokerage/transfer fees; as long as these are below Rs 2,00,000, the arbitrage is profitable.

How arbitrage eliminates its own opportunity

The profit opportunity does not survive for long, because arbitrage itself changes the prices that created it:

1. Buying pressure in the cheap market — As arbitrageurs rush to buy USD in Kathmandu, demand rises and the Kathmandu rate is pushed up (say from Rs 130 toward Rs 131).
2. Selling pressure in the dear market — As they sell USD in New York, supply rises and the New York rate is pushed down (say from Rs 132 toward Rs 131).
3. Convergence of prices — The two rates move toward each other. Once they become equal (Rs 131 = Rs 131), the price difference disappears and the profit margin falls to zero.
4. Entry of many arbitrageurs — Because arbitrage is riskless, many traders enter at once. Their combined volume accelerates the price adjustment, so the window closes very quickly — often within seconds in modern electronic markets.

This self-correcting mechanism is the essence of the Law of One Price: identical assets must sell for the same price in efficient markets. Arbitrage is the force that enforces this law. Once prices converge, arbitrageurs stop trading (there is nothing left to gain), and the market returns to equilibrium. Hence arbitrage is described as a self-eliminating activity — the very act of exploiting a price gap destroys the gap.

Given mean
[ \mu=\overline{S}(n)=3.0 ] and variance
[ \sigma^2=3 ]

The Gaussian (normal) probability density function is
[ f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right] ]

At the mean value (x=\mu=3.0),
[ f(3)=\frac{1}{\sqrt{2\pi(3)}}\exp(0) ]

[ f(3)=\frac{1}{\sqrt{6\pi}} ]

[ f(3)\approx \frac{1}{4.3416}=0.2303 ]

[ \boxed{f(3)\approx 0.2303} ]

So, the value of the Gaussian probability distribution with variance (3) at its mean (3.0) is approximately 0.2303.

Proof of the characteristic function

Let (X\sim \text{Gamma}(\gamma,1)), (\gamma>0), with density

[ f(x)= \begin{cases} \dfrac{x^{\gamma-1}e^{-x}}{\Gamma(\gamma)}, & x>0,\[6pt] 0, & x\le 0. \end{cases} ]

The characteristic function of (X) is

[ \phi(q)=E(e^{iqX})=\int_0^\infty e^{iqx}f(x),dx. ]

So

[ \phi(q)=\frac{1}{\Gamma(\gamma)}\int_0^\infty x^{\gamma-1}e^{-(1-iq)x},dx. ]

Since (q) is real, (\operatorname{Re}(1-iq)=1>0), so the integral converges. Put

[ t=(1-iq)x,\qquad dx=\frac{dt}{1-iq}. ]

Then

[ \phi(q) =\frac{1}{\Gamma(\gamma)}\int_0^\infty \left(\frac{t}{1-iq}\right)^{\gamma-1}e^{-t}\frac{dt}{1-iq}. ]

Hence

[ \phi(q) =\frac{1}{\Gamma(\gamma)(1-iq)^\gamma} \int_0^\infty t^{\gamma-1}e^{-t},dt. ]

But

[ \int_0^\infty t^{\gamma-1}e^{-t},dt=\Gamma(\gamma). ]

Therefore

[ \phi(q)=\frac{\Gamma(\gamma)}{\Gamma(\gamma)(1-iq)^\gamma} =(1-iq)^{-\gamma}. ]

Thus the characteristic function of the Gamma distribution is

[ \boxed{\phi(q)=(1-iq)^{-\gamma}},\qquad \gamma>0. ]

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Infinite divisibility

Yes, the Gamma distribution is infinitely divisible.

A distribution is infinitely divisible if for every positive integer (n), there exist independent and identically distributed random variables (Y_1,Y_2,\dots,Y_n) such that

[ Y_1+Y_2+\cdots+Y_n ]

has the same distribution as (X).

Let

[ Y_k\sim \text{Gamma}\left(\frac{\gamma}{n},1\right),\qquad k=1,2,\dots,n, ]

and let (Y_1,Y_2,\dots,Y_n) be independent. The characteristic function of each (Y_k) is

[ \phi_{Y_k}(q)=\left(1-iq\right)^{-\gamma/n}. ]

Hence the characteristic function of

[ S_n=Y_1+Y_2+\cdots+Y_n ]

is

[ \phi_{S_n}(q) =\prod_{k=1}^n \phi_{Y_k}(q) =\left[\left(1-iq\right)^{-\gamma/n}\right]^n =(1-iq)^{-\gamma}. ]

But this is exactly the characteristic function of (X\sim \text{Gamma}(\gamma,1)). By the uniqueness theorem of characteristic functions,

[ S_n \overset{d}{=} X. ]

Thus for every (n), (X) can be expressed as the sum of (n) iid Gamma(\left(\frac{\gamma}{n},1\right)) random variables. Therefore,

[ \boxed{\text{The Gamma distribution is infinitely divisible.}} ]

Equivalently, since the sum of independent Gamma variables with the same scale is again Gamma with shape equal to the sum of the shapes, we have

[ \text{Gamma}\left(\frac{\gamma}{n},1\right)+\cdots+\text{Gamma}\left(\frac{\gamma}{n},1\right) \sim \text{Gamma}(\gamma,1). ]

Hence the Gamma distribution is infinitely divisible.

Part A — Physics methods in socio‑economic problems

Idea. Socio‑economic systems are made of a very large number of interacting units (people, firms, banks, cities) whose individual behaviour is simple but whose collective behaviour is not. This is exactly the situation statistical physics was built to handle. The field that applies physics tools to economics and society is called econophysics.

1. Phase transitions

A phase transition occurs when a control parameter crosses a critical value $T_c$ and the system changes qualitatively. Near $T_c$ the order parameter (a measure of collective order) vanishes as a power law, correlations become long‑ranged, and fluctuations diverge. Social systems show the same mathematics.

• Opinion dynamics (voter / Ising model). Let each agent hold opinion $\pm 1$ and let $J$ be the "social pressure" to conform. Below a critical conformity $J_c$ opinions stay disordered (order parameter $m = 0$); above it a consensus forms ($m \neq 0$). This is a symmetry‑breaking transition, and near $J_c$ the magnetization behaves as $m \propto (J_c - J)^{\beta}$.
• Schelling segregation model. Even when every individual is only mildly tolerant (wants, say, 30 % of neighbours like himself), the city undergoes a sharp transition from an integrated to a strongly segregated state. A small change in tolerance produces a large, discontinuous change in the social pattern.
• Financial crashes. Sornette's log‑periodic power law describes bubbles as critical phenomena: the price and the crash risk accelerate as $|t_c - t|^{-m}$ with superimposed oscillations as the critical time $t_c$ is approached. The 1929 and 2000 crashes fit this form.
• Emergence of money / conventions. Models of trading show a phase transition from barter to a single accepted medium of exchange as the number of goods or the memory of agents crosses a threshold.

2. Non‑linear dynamics and chaos

Economic variables feed back on themselves, so the equations are non‑linear. Such systems show bifurcations, limit cycles, chaos (sensitive dependence on initial conditions) and synchronization.

• Cobweb model of farm prices. Farmers plant according to last year's price; supply responds with a lag. The price sequence obeys a non‑linear map $p_{t+1} = f(p_t)$ which can settle to a fixed point, oscillate in a limit cycle, or become chaotic — explaining irregular agricultural price cycles.
• Business cycles. Multiplier–accelerator models produce limit‑cycle oscillations; adding non‑linearity gives chaotic fluctuations without any external shock.
• Chaotic exchange rates and stock indices. Measured Lyapunov exponents of some financial time series are positive, indicating deterministic chaos rather than pure randomness.
• Synchronization of economies. Coupled oscillators (trade partners) can lock into a common cycle — the physics of coupled pendulums applied to global business cycles.

3. Disorder systems (spin glasses and frustration)

A spin glass has quenched random interactions and frustration — no single configuration satisfies all bonds. Its energy landscape is rugged, with many nearly equal valleys. This is exactly the situation in many social and economic problems.

• Portfolio optimization. Choosing assets to maximise return and minimise risk is analogous to finding the ground state of a spin glass; the Sherrington–Kirkpatrick model and replica methods give the optimal portfolio and its phase diagram.
• Combinatorial optimization. Problems such as the travelling salesman, graph partitioning and job scheduling map onto disordered magnets; simulated annealing, borrowed from statistical mechanics, is a standard tool.
• Financial markets. The Sornette–Zhou and Bornholdt models treat traders as spins with random couplings; market crashes appear as a glassy transition from a "hot" disordered phase to a "frozen" ordered phase.
• Neural networks and memory. Hopfield's model of associative memory is a disordered spin system; its storage capacity is a spin‑glass phase transition.

4. Other physics areas used

• Network theory. Social, trade and banking networks are scale‑free and small‑world. Percolation theory gives the epidemic threshold $\lambda_c = \langle k \rangle / \langle k^2 \rangle$; the same idea explains how the failure of one bank can cascade through the interbank network (2008 crisis).
• Statistical mechanics of money. In the Dragulescu–Yakovenko model, random pairwise transfers of money give a Gibbs (exponential) distribution for the middle class and a Pareto power law for the rich — the observed two‑tailed wealth distribution.
• Random matrix theory. Laloux, Bouchaud, Potters and co‑workers showed that most eigenvalues of stock‑return correlation matrices match the random‑matrix (Wigner) prediction, while a few large eigenvalues carry genuine market‑wide information. This is used to build better risk models.
• Self‑organized criticality. Sandpile, forest‑fire and earthquake models show systems that drive themselves to a critical state, producing power‑law avalanches. Stock‑market crashes and firm‑size fluctuations follow the same statistics.
• Scaling and fractals. Zipf's law for city sizes, Pareto law for incomes, and Hurst‑exponent (fractal) analysis of price series all come from physics of scale invariance.

Conclusion. Phase transitions, non‑linear dynamics, disordered systems, percolation, networks and scaling give a common language for problems as different as consensus formation, market crashes, city growth and income inequality. The same order parameters, critical exponents and power laws appear in both physics and society.

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Part B — The St. Petersburg Paradox

Statement. A fair coin is tossed until the first head appears. If the head appears on the $n$-th toss, the player receives $x = 2^n$ rupees. What is a fair price to play this game?

Expected payoff.

$E[X] = \sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n 2^n = \sum_{n=1}^{\infty} 1 = 1+1+1+\cdots = \infty$

The expected value is infinite, so a "fair" entry fee would also be infinite. Yet in practice nobody pays more than a few rupees. This contradiction between theory and behaviour is the St. Petersburg Paradox (Nicolaus Bernoulli, resolved by Daniel Bernoulli, 1738).

Formulation as a power-law distribution.

The payoff is $x = 2^n$ and its probability is

$P(X = x) = \left(\frac{1}{2}\right)^n = \frac{1}{x} = x^{-1}.$

So the payoff follows a power law $P(x) \propto x^{-\alpha}$ with exponent $\alpha = 1$ (on the geometric grid $x = 2, 4, 8, \dots$). The tail (survival) function follows by summing the geometric series:

$P(X \ge 2^n) = \sum_{m \ge n} 2^{-m} = 2^{-n+1} = \frac{2}{x}, \qquad\text{i.e. } S(x) \propto x^{-1}.$

Writing the survival function as $S(x) \propto x^{-\mu}$ (tail exponent $\mu$), the St. Petersburg game has $\mu = 1$. In terms of a continuous density this is $f(x) \propto x^{-(\mu+1)} = x^{-2}$.

Moments. For $f(x)\propto x^{-(\mu+1)}$ the $k$-th moment $\int x^k f(x)\,dx$ converges only if $k < \mu$. Hence:

• the mean ($k=1$) needs $\mu > 1$; here $\mu = 1$, so the mean diverges (marginally, like the harmonic series);
• the variance ($k=2$) needs $\mu > 2$, so it diverges too.

Interpretation. The paradox is not a failure of probability theory but a property of a heavy-tailed (power-law) distribution with a divergent first moment. Rare, enormous payoffs (long runs of tails) carry so much weight that they dominate the average. The same mathematics appears in socio-economic data. Using the survival-function exponent $\mu$ (the Pareto exponent), real income and wealth distributions have $\mu \approx 1.5$–$2$: the mean is finite ($\mu>1$) but the variance is infinite ($\mu \le 2$). Zipf's law for city sizes has $\mu \approx 1$, exactly on the borderline of the St. Petersburg game.

Resolutions.

1. Diminishing marginal utility (Bernoulli). People maximise expected utility, not money. With $u(x) = \ln x$,

$E[u] = \sum_{n=1}^{\infty} 2^{-n}\ln 2^n = \ln 2 \sum_{n=1}^{\infty} \frac{n}{2^n} = 2\ln 2,$

which is finite, so a finite fee is justified. Any utility growing slower than linearly (e.g. $u \propto \sqrt{x}$ or bounded utility) removes the paradox.

2. Finite resources of the casino. A real bank cannot pay $2^n$ for arbitrarily large $n$; if the game is truncated at $n = N$ tosses, the expected value is only $N$.

3. Finite time: any real game has a finite number of tosses.

4. Risk aversion: the huge but extremely improbable payoff is not worth a large certain cost.

Question: Consider (n) independent identically distributed random variables in a random walk. Obtain the expectation value of the sum (S_n) and its variance.

Setup

Let (X_1, X_2, \dots, X_n) be (n) independent and identically distributed (i.i.d.) random variables. In a random walk, each (X_i) represents the displacement at the (i)-th step. The position after (n) steps is the sum

$S_n = X_1 + X_2 + \cdots + X_n = \sum_{i=1}^{n} X_i$

Let the common mean and variance of each (X_i) be

$E(X_i) = \mu, \qquad \operatorname{Var}(X_i) = \sigma^2$

for every (i = 1, 2, \dots, n).

1. Expectation value of (S_n)

Using the linearity of expectation,

$E(S_n) = E\left(\sum_{i=1}^{n} X_i\right) = \sum_{i=1}^{n} E(X_i)$

Since the variables are identically distributed with (E(X_i) = \mu),

$E(S_n) = \sum_{i=1}^{n} \mu = n\mu$

$\boxed{E(S_n) = n\mu}$

Thus the expected position after (n) steps is (n) times the mean displacement per step. For a symmetric random walk with (\mu = 0), we get (E(S_n) = 0).

2. Variance of (S_n)

By definition,

$\operatorname{Var}(S_n) = E\left[\left(S_n - E(S_n)\right)^2\right] = E\left[\left(\sum_{i=1}^{n} (X_i - \mu)\right)^2\right]$

Expanding the square,

$\operatorname{Var}(S_n) = \sum_{i=1}^{n} E\left[(X_i - \mu)^2\right] + 2\sum_{i<j} E\left[(X_i - \mu)(X_j - \mu)\right]$

Now, because the (X_i) are independent, for (i \neq j),

$E\left[(X_i - \mu)(X_j - \mu)\right] = E(X_i - \mu)\,E(X_j - \mu) = 0 \cdot 0 = 0$

So all the cross-covariance terms vanish. Also,

$E\left[(X_i - \mu)^2\right] = \operatorname{Var}(X_i) = \sigma^2$

Therefore,

$\operatorname{Var}(S_n) = \sum_{i=1}^{n} \sigma^2 = n\sigma^2$

$\boxed{\operatorname{Var}(S_n) = n\sigma^2}$

The standard deviation is therefore

$\sigma_{S_n} = \sqrt{n}\,\sigma$

which grows as the square root of the number of steps — the characteristic (\sqrt{n}) behaviour of a random walk.

Summary

For (n) i.i.d. random variables with mean (\mu) and variance (\sigma^2):

$E(S_n) = n\mu, \qquad \operatorname{Var}(S_n) = n\sigma^2$

Note: The independence of the (X_i) is essential for the variance result, since it makes every covariance term zero. If the variables were correlated, the variance would contain additional covariance terms.

Stochastic Process

A stochastic process is a collection of random variables $\{X(t) : t \in T\}$ indexed by a parameter $t$ (usually time), defined on a common probability space. For each fixed $t$, $X(t)$ is a random variable; for each fixed outcome (sample point) $\omega$, the function $t \mapsto X(t,\omega)$ is called a sample path or realization of the process.

In other words, a stochastic process describes the evolution of some random quantity over time. The index set $T$ may be discrete (e.g. $T = \{0,1,2,\dots\}$, giving a discrete-time process or time series) or continuous (e.g. $T = [0,\infty)$).

Examples: daily closing price of a stock, the number of customers in a queue at time $t$, the Wiener (Brownian motion) process, and a Markov chain.

A stochastic process is characterized by its finite-dimensional distributions, i.e. the joint distribution of $(X(t_1), X(t_2), \dots, X(t_n))$ for any $n$ and any times $t_1 < t_2 < \dots < t_n$.

Stationary Stochastic Process

A process is stationary when its probabilistic behaviour does not change with a shift in the time origin. There are two notions:

1. Strictly stationary (strongly stationary): A process $\{X(t)\}$ is strictly stationary if for every $n$, every set of times $t_1, \dots, t_n$, and every shift $h$, the joint distribution is invariant under time translation: $F_{X(t_1),\dots,X(t_n)}(x_1,\dots,x_n) = F_{X(t_1+h),\dots,X(t_n+h)}(x_1,\dots,x_n).$ That is, the joint distribution depends only on the time differences, not on the actual time.

2. Weakly stationary (second-order / covariance stationary): A process is weakly stationary if:

• the mean is constant: $\mathbb{E}[X(t)] = \mu$ for all $t$;
• the variance is finite and constant: $\operatorname{Var}(X(t)) = \sigma^2 < \infty$;
• the autocovariance depends only on the lag $h = t_2 - t_1$: $\operatorname{Cov}(X(t), X(t+h)) = \gamma(h).$

Properties and remarks:

• Strict stationarity with finite second moments implies weak stationarity; the converse is not generally true.
• For a Gaussian process, weak stationarity implies strict stationarity, since the normal distribution is fully determined by its mean and covariance.
• The autocorrelation function $\rho(h) = \gamma(h)/\gamma(0)$ satisfies $\rho(0)=1$ and $|\rho(h)| \le 1$.
• Stationarity is essential in time-series analysis because it allows the past to be used to predict the future; non-stationary series (with trend or seasonality) must first be transformed (e.g. by differencing) into stationary ones.

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OR

Time and Price Scales in Financial Markets

In financial markets, prices are recorded against time, and the way we choose the time scale and the price scale strongly affects the statistical properties of the resulting series. These two scales form the framework for modelling asset prices, volatility, and market microstructure.

1. Time Scales

Time is the horizontal axis along which market activity is measured. Several choices exist:

• Calendar time (physical time): Fixed intervals such as one day, one week, or one month. Most financial data (daily closing prices) are recorded this way. However, calendar time ignores that markets are closed at night and on holidays, so returns over equal calendar intervals do not correspond to equal amounts of trading activity.
• Trading time (market time): Time measured only when the market is open. This removes the distortion caused by overnight and weekend gaps and is more natural for intraday analysis.
• Transaction (event) time: Time advances only when a trade occurs. The series is then indexed by trade number rather than by clock time. This "event-based" scale makes the spacing between observations irregular but aligns observations with actual market activity.
• Intrinsic time / volatility time: Time is measured by the cumulative variance (or by number of trades/volume). Under this scale, returns become closer to normal and volatility is more stable — an idea used in volatility clocks and in the "business time" of market microstructure models.

Importance: The choice of time scale affects estimated volatility, autocorrelation, and the observed distribution of returns. Returns computed over finer time scales show heavier tails and stronger autocorrelation of volatility; as the time scale increases, returns tend to approach normality (aggregation effect).

2. Price Scales

Price is the vertical axis. The relevant choices are:

• Linear (arithmetic) price scale: Equal vertical distances represent equal absolute price changes. Simple returns $R_t = (P_t - P_{t-1})/P_{t-1}$ are natural here. Suitable for short horizons and for assets whose price changes are small relative to the level.
• Logarithmic (log) price scale: Equal vertical distances represent equal proportional (percentage) changes. Log prices $p_t = \ln P_t$ are used, giving log returns $r_t = \ln(P_t/P_{t-1})$. This scale is preferred because:
  • it makes proportional changes comparable across different price levels and across time;
  • log returns are approximately equal to simple returns for small changes and are additive over time: $r_{t_1,t_2} = r_{t_1} + \dots + r_{t_2}$;
  • it is consistent with the geometric Brownian motion model of asset prices, where log returns are normally distributed.
• Tick size and price discreteness: In real markets, prices move in minimum increments (ticks), so the price scale is discrete. This discreteness matters at very fine time scales (bid–ask spread, rounding) but becomes negligible at coarser scales.

3. Relationship Between the Two Scales

Time and price scales interact. On a fine time scale, price changes are dominated by microstructure noise (discreteness, bid–ask bounce); on a coarse time scale, they reflect fundamental movements. Similarly, using log prices with calendar time gives the standard return series used in most financial models, while using transaction time with log prices gives a series closer to a random walk with stable volatility.

Conclusion: The time scale determines when we observe prices (calendar, trading, or event time), and the price scale determines how we measure changes (linear or logarithmic). A proper choice of both scales is essential for correct estimation of returns, volatility, and for building realistic models of financial markets.

a) Brief note on algorithmic complexity theory

Algorithmic complexity theory (also called computational complexity theory) studies how the time and memory required by an algorithm grow as the size of its input grows. It gives a mathematical way to compare algorithms independently of the machine on which they run.

The input size is denoted by $n$. The running time is expressed using asymptotic notation:

• Big-O, $O(f(n))$ — an upper bound: the time grows no faster than $f(n)$.
• Big-Omega, $\Omega(f(n))$ — a lower bound: the time grows at least as fast as $f(n)$.
• Big-Theta, $\Theta(f(n))$ — a tight bound: the time grows exactly at the rate of $f(n)$.

Common complexity classes, in increasing order of growth, are:

$O(1) < O(\log n) < O(n) < O(n\log n) < O(n^2) < O(2^n) < O(n!)$

For example, linear search on $n$ elements takes $O(n)$ time in the worst case, binary search takes $O(\log n)$, and bubble sort takes $O(n^2)$. Problems that can be solved in polynomial time belong to class P; those whose solutions can be verified in polynomial time belong to class NP. The famous P vs NP question asks whether these two classes are equal and remains unsolved.

Thus, complexity theory provides a standard framework for measuring the efficiency of algorithms and for classifying problems by their inherent difficulty.

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b) Meaning of volatility

Volatility refers to the degree of variation or fluctuation in a variable over time. In statistics and finance, it measures how rapidly and how widely the values of a series (such as stock prices, exchange rates or returns) change from one period to the next.

Key points:

• High volatility means values change sharply and frequently; the series is unstable and unpredictable.
• Low volatility means values remain relatively steady and move within a narrow range.
• It is commonly measured by the standard deviation of returns. If $r_t$ denotes the return at time $t$ and $\bar{r}$ the mean return, then

$\sigma = \sqrt{\frac{1}{n-1}\sum_{t=1}^{n}(r_t-\bar{r})^2}$

• In time series analysis, volatility is often studied through conditional variance models such as ARCH and GARCH, where the variance of the present period depends on past squared errors.
• Volatility is used in finance for risk assessment, option pricing (e.g. the Black–Scholes model) and portfolio management. A highly volatile asset is considered riskier because its future value is more uncertain.

In short, volatility is a measure of the extent of uncertainty or dispersion in a time series, indicating how much the values are expected to deviate from their average level.

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c) White noise

Definition: A time series $\{e_t\}$ is called white noise if it is a sequence of uncorrelated random variables having constant mean and constant variance, with no autocorrelation at any lag. Formally, $\{e_t\}$ is white noise if:

1. $E(e_t) = 0$ for all $t$ (zero mean),
2. $\text{Var}(e_t) = \sigma^2$ for all $t$ (constant variance),
3. $\text{Cov}(e_t, e_{t-k}) = 0$ for all $k \neq 0$ (no autocorrelation).

The autocorrelation function is therefore

$\rho_k = \begin{cases} 1, & k = 0 \\ 0, & k \neq 0 \end{cases}$

If, in addition, the variables are independent and normally distributed, it is called Gaussian white noise, written $e_t \sim N(0,\sigma^2)$.

Example: Tossing a fair coin repeatedly and recording the outcome as $+1$ for a head and $-1$ for a tail gives a white noise series. Each toss is independent of the others, the mean is zero, the variance is constant, and there is no correlation between successive tosses. Similarly, the residuals obtained after fitting a good time series model should behave like white noise; if they do not, the model is inadequate.

Statement: "The market doesn't have memory" means that in an efficient market, past prices and past events carry no predictive power over future prices. Each day's price is formed fresh from today's demand and supply forces; yesterday's price level leaves no trace that can be systematically exploited. In other words, the market is memoryless — it does not "remember" whether it rose or fell previously, so past data cannot be used to reliably predict the next move. This is the essence of the weak-form Efficient Market Hypothesis (EMH) and the Random Walk Hypothesis.

Justification:

1. Random walk behaviour: If prices follow a random walk, the change from today to tomorrow is independent of the change from yesterday to today. The correlation between successive price changes is statistically zero, so history gives no edge.

2. Rapid information absorption: As soon as new information arrives, it is instantly and fully reflected in the price. Once absorbed, the information is already "priced in," so it cannot be exploited again — the market "forgets" it in the sense that it offers no further profit opportunity.

3. Competition among investors: Thousands of rational, profit-seeking traders compete to exploit any predictable pattern. The moment a pattern is noticed, trading activity eliminates it, restoring randomness.

4. No arbitrage profit from past data: If the market had memory (e.g., "prices always rise after three falls"), investors would buy on that signal, pushing prices up immediately and destroying the pattern.

Example with graph:

Suppose a stock's price over time is plotted. Under the random walk, the series shows no repeating cycle or trend — each move is a fresh, unpredictable step.

Price (Rs)
  |
120|                    *        *
   |          *     *        *
100|     *        *        *
   |  *     *        *
 80|*   *        *
   |___________________________ Time (days)
    1  2  3  4  5  6  7  8  9 10

The path wanders up and down irregularly. A rise from day 1 to day 2 tells us nothing about day 3; a fall from day 4 to day 5 does not guarantee a rise on day 6. There is no discernible pattern — the graph confirms that past movements do not determine future movements.

Conclusion: Since future price changes are independent of past changes and all available information is already reflected in current prices, the market has no memory. Investors cannot earn abnormal returns by studying historical price patterns alone.

Given: Step size, (l = 1.5\ \text{ft} = 1.5 \times 0.3048 = 0.4572\ \text{m}) Time step, (\tau = 0.9\ \text{s})

For a one-dimensional random walk, the diffusion constant is [ D = \frac{l^2}{2\tau} ]

Substituting the values: [ D = \frac{(0.4572)^2}{2 \times 0.9} = \frac{0.2090}{1.8} = 0.1161\ \text{m}^2\text{s}^{-1} ]

In feet units: [ D = \frac{(1.5)^2}{2 \times 0.9} = \frac{2.25}{1.8} = 1.25\ \text{ft}^2\text{s}^{-1} ]

[ \boxed{D \approx 0.116\ \text{m}^2/\text{s} \approx 1.25\ \text{ft}^2/\text{s}} ]

Note: If the random walk is three-dimensional, the relation is (\langle r^2 \rangle = 6Dt), so [ D = \frac{l^2}{6\tau} = \frac{(0.4572)^2}{6 \times 0.9} = 0.0387\ \text{m}^2/\text{s} ]

Characteristic function of the Poisson distribution

Let $X \sim \text{Poisson}(\lambda)$, so that its probability mass function is

$P(X = x) = \frac{e^{-\lambda}\lambda^{x}}{x!}, \qquad x = 0, 1, 2, \dots$

The characteristic function of $X$ is defined as

$\phi(q) = E\left(e^{iqX}\right) = \sum_{x=0}^{\infty} e^{iqx}\, P(X = x)$

Substituting the pmf,

$\phi(q) = \sum_{x=0}^{\infty} e^{iqx}\, \frac{e^{-\lambda}\lambda^{x}}{x!} = e^{-\lambda} \sum_{x=0}^{\infty} \frac{\left(\lambda e^{iq}\right)^{x}}{x!}$

Using the exponential series $\displaystyle\sum_{x=0}^{\infty} \frac{z^{x}}{x!} = e^{z}$ with $z = \lambda e^{iq}$,

$\phi(q) = e^{-\lambda}\, e^{\lambda e^{iq}} = e^{\lambda e^{iq} - \lambda}$

Hence

$\boxed{\phi(q) = e^{\lambda\left(e^{iq} - 1\right)}}$

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Infinite divisibility

A random variable $X$ (or its characteristic function) is said to be infinitely divisible if for every positive integer $n$ there exist independent and identically distributed random variables $X_1, X_2, \dots, X_n$ such that

$X \stackrel{d}{=} X_1 + X_2 + \cdots + X_n$

Equivalently, the characteristic function must satisfy

$\phi(q) = \left[\phi_n(q)\right]^{n}$

for every $n$, where $\phi_n(q)$ is itself a valid characteristic function.

Verification for the Poisson distribution

Fix any positive integer $n$ and define

$\phi_n(q) = e^{\frac{\lambda}{n}\left(e^{iq} - 1\right)}$

This is exactly the characteristic function of a Poisson random variable with parameter $\lambda/n$. Therefore $\phi_n(q)$ is a genuine characteristic function (it corresponds to the $\text{Poisson}(\lambda/n)$ law).

Now raise it to the power $n$:

$\left[\phi_n(q)\right]^{n} = \left[e^{\frac{\lambda}{n}\left(e^{iq} - 1\right)}\right]^{n} = e^{\lambda\left(e^{iq} - 1\right)} = \phi(q)$

Thus for every $n$ we can write

$\phi(q) = \left[\phi_n(q)\right]^{n}, \qquad \phi_n(q) = e^{\frac{\lambda}{n}\left(e^{iq}-1\right)}$

Since each $\phi_n(q)$ is the characteristic function of an independent $\text{Poisson}(\lambda/n)$ variable, we have

$X \stackrel{d}{=} X_1 + X_2 + \cdots + X_n, \qquad X_i \stackrel{\text{iid}}{\sim} \text{Poisson}\!\left(\frac{\lambda}{n}\right)$

This holds for every positive integer $n$. Hence the Poisson distribution is infinitely divisible.

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Conclusion

The characteristic function of the Poisson distribution is

$\phi(q) = e^{\lambda\left(e^{iq} - 1\right)}$

and since its $n$-th root $e^{\frac{\lambda}{n}(e^{iq}-1)}$ is again the characteristic function of a Poisson$(\lambda/n)$ variable for every $n$, the Poisson process is infinitely divisible.