Econophysics

Econophysics is an interdisciplinary research field that applies theories, methods, and computational tools originally developed in physics (especially statistical mechanics, thermodynamics, and the physics of complex systems) to the study of economic and financial phenomena. The term was coined around 1995 (popularized by H. Eugene Stanley and others) and has since grown into a major area of interdisciplinary science.

The core idea is that an economy is a complex system made up of a very large number of interacting agents (firms, traders, consumers), much like a gas is made up of a very large number of interacting molecules. Therefore, the mathematical machinery physicists use to describe many-particle systems can be transferred, with suitable modification, to describe economic systems.

Main themes of econophysics

1. Statistical mechanics of markets — treating price movements as stochastic processes and studying their probability distributions.
2. Power laws and scaling — many economic quantities (income, wealth, firm size, price fluctuations) follow power-law distributions rather than normal distributions.
3. Financial markets as complex systems — studying volatility clustering, fat tails, long-range correlations, and crashes.
4. Agent-based modelling — simulating markets as collections of interacting heterogeneous agents.
5. Networks — modelling trade, banking, and ownership as networks to study systemic risk.

Why physicists are motivated to do interdisciplinary research in econophysics

Physicists are motivated for several reasons:

1. Universality of physical laws. Physicists have discovered that many systems, though microscopically different, obey the same macroscopic laws (universality classes). They suspect economic systems may show similar universal behaviour, so the same mathematical tools may apply.

2. Availability of huge data. Modern financial markets generate enormous amounts of high-frequency data. Physicists are trained to analyse large data sets and extract statistical regularities, so financial data is an attractive testing ground.

3. Complex systems are a natural physics problem. The economy is a paradigmatic complex system with many interacting components, feedback, and emergent behaviour — exactly the kind of system statistical physicists study.

4. Failure of standard economic assumptions. Traditional economics often assumes rational agents and normal (Gaussian) distributions. Empirical data show fat tails, irrational behaviour, and non-equilibrium dynamics. Physicists offer alternative, empirically grounded models.

5. Interdisciplinary appeal and new phenomena. Econophysics provides new problems (market crashes, bubbles, wealth condensation) that challenge physical intuition and extend physics itself.

6. Practical and financial interest. Banks, hedge funds, and regulators increasingly employ physicists ("quants") for risk management, option pricing, and portfolio optimization. This creates strong career and funding incentives.

7. Methodological transfer. Concepts such as entropy, temperature, free energy, phase transitions, percolation, and scaling laws have direct analogues in economics (e.g., market "temperature", "critical" crashes), making the transfer intellectually rewarding.

Thus econophysics is a genuine two-way bridge: physics gains new testable systems, and economics gains powerful quantitative tools for understanding markets and the macroeconomy.

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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 (or mean) of many independent 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 error of the normal approximation.

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

[ E[X_i] = \mu, \qquad \operatorname{Var}(X_i) = \sigma^2 > 0, \qquad \rho = E\big[|X_i - \mu|^3\big] < \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 (\Phi(x)) the CDF of the standard normal distribution.

Berry–Esseen Theorem I (Uniform bound)

There exists a universal constant (C) (independent of (n) and of the distribution) such that for all (x \in \mathbb{R}),

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

This is the uniform (Kolmogorov-type) version: the maximum deviation between the true CDF and the normal CDF is bounded by a constant times (1/\sqrt{n}).

Berry–Esseen Theorem II (Non-uniform / local version)

A sharper, pointwise version states that for every (x),

[ \big| F_n(x) - \Phi(x) \big| \le \frac{C',\rho}{\sigma^3 \sqrt{n}} \cdot \frac{1}{1 + |x|^3}, ]

where (C') is another universal constant. This is the non-uniform (or local) Berry–Esseen bound. It shows that the error is not only (O(1/\sqrt{n})) uniformly, but actually decays faster in the tails — the normal approximation is extremely good far out in the tails, where the factor (1/(1+|x|^3)) becomes very small.

Statement in words

• Theorem I: The largest possible gap between the distribution of the standardized sum and the standard normal distribution is at most of order (1/\sqrt{n}), with the constant depending only on the third absolute moment.
• Theorem II: In the tails ((|x|) large), this gap is even smaller, of order (1/(|x|^3\sqrt{n})).

Importance of the Berry–Esseen theorem

1. Rate of convergence. It supplies the missing "how fast" in the CLT, giving an explicit (O(1/\sqrt{n})) error rate.
2. Error control in practice. In statistics, finance, and physics, one often replaces a sum of random variables by a normal variable. The theorem tells us how large the resulting error can be, so we know how large (n) must be for a desired accuracy.
3. Dependence on moments. The bound involves (\rho/\sigma^3) (a measure of skewness and kurtosis), so it shows that distributions close to normal converge faster, while highly skewed or heavy-tailed distributions converge more slowly.
4. Sharpness. The (1/\sqrt{n}) rate is optimal in general — no better uniform rate is possible — so the theorem is essentially the best possible result of its kind.
5. Foundation for further results. It underlies Edgeworth expansions, large-deviation estimates, and many results in asymptotic statistics, econometrics, and risk theory.

Central Limit Theorem: Derivation, Conditions, and Significance in Finance

1. Statement of the Central Limit Theorem (CLT)

Let $X_1, X_2, \ldots, X_n$ be a sequence of independent and identically distributed (i.i.d.) random variables, each with finite mean $\mu$ and finite variance $\sigma^2$. Define the sample mean

$\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n} X_i$

Then, as $n \to \infty$, the standardized variable

$Z_n = \frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} = \frac{\sum_{i=1}^{n} X_i - n\mu}{\sigma\sqrt{n}}$

converges in distribution to the standard normal distribution:

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

Equivalently, for large $n$,

$\bar{X}_n \sim N\!\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{and} \quad \sum_{i=1}^{n} X_i \sim N\!\left(n\mu, n\sigma^2\right)$

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2. Derivation Using Moment Generating Functions

Step 1: Standardize each observation.

Let

$Y_i = \frac{X_i - \mu}{\sigma}$

Then $E(Y_i) = 0$ and $\text{Var}(Y_i) = 1$. The standardized sum becomes

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

Step 2: Write the moment generating function (MGF) of $Z_n$.

Since the $Y_i$ are independent and identically distributed,

$M_{Z_n}(t) = E\!\left(e^{t Z_n}\right) = E\!\left(\exp\!\left(\frac{t}{\sqrt{n}}\sum_{i=1}^{n} Y_i\right)\right)$

$= \prod_{i=1}^{n} E\!\left(e^{t Y_i/\sqrt{n}}\right) = \left[M_Y\!\left(\frac{t}{\sqrt{n}}\right)\right]^{n}$

Step 3: Expand the MGF of $Y$ as a Taylor series about 0.

Since $E(Y) = 0$ and $E(Y^2) = 1$,

$M_Y(s) = 1 + s\,E(Y) + \frac{s^2}{2!}E(Y^2) + \frac{s^3}{3!}E(Y^3) + \cdots$

$= 1 + \frac{s^2}{2} + O(s^3)$

Step 4: Substitute $s = t/\sqrt{n}$.

$M_Y\!\left(\frac{t}{\sqrt{n}}\right) = 1 + \frac{t^2}{2n} + O\!\left(n^{-3/2}\right)$

Step 5: Raise to the power $n$ and take the limit.

$M_{Z_n}(t) = \left[1 + \frac{t^2}{2n} + O\!\left(n^{-3/2}\right)\right]^{n}$

Using the standard limit $\lim_{n\to\infty}\left(1 + \frac{a}{n}\right)^{n} = e^{a}$ with $a = t^2/2$,

$\lim_{n\to\infty} M_{Z_n}(t) = e^{t^2/2}$

Step 6: Recognize the limiting MGF.

The function $e^{t^2/2}$ is the MGF of the standard normal distribution $N(0,1)$. By the uniqueness theorem of MGFs, convergence of MGFs implies convergence in distribution:

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

This completes the derivation of the CLT.

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3. Essential Conditions of the CLT

1. Independence — The observations $X_1, X_2, \ldots, X_n$ must be independent of one another.
2. Identical distribution — All observations must come from the same probability distribution (i.i.d. assumption).
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. Sufficiently large sample size — Generally $n \ge 30$ is considered adequate; the more skewed the population, the larger the $n$ required.
5. Random sampling — The sample must be drawn randomly so that each observation is representative of the population.

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4. Significance of the CLT in Finance

The CLT is a cornerstone of modern financial theory and practice for the following reasons:

a) Portfolio diversification and risk measurement. The return on a diversified portfolio is essentially a weighted average of many individual asset returns. By the CLT, portfolio returns tend to be approximately normally distributed, which justifies using mean–variance analysis (Markowitz) and standard deviation as a measure of risk.

b) Value at Risk (VaR) and option pricing. VaR models and the Black–Scholes option pricing formula rely on the assumption that asset returns are normally distributed. The CLT provides the theoretical justification for this normality assumption when returns result from many small independent influences.

c) Sampling and estimation. Financial analysts rarely observe the entire population of returns. The CLT allows them to use sample means and standard errors to construct confidence intervals and conduct hypothesis tests about average returns, volatility, and other parameters.

d) Aggregation of risks. Insurance and banking institutions pool thousands of independent risks (loans, policies). The CLT ensures that the aggregate loss is approximately normal, enabling accurate calculation of reserves, capital requirements, and premiums.

e) Monte Carlo simulation. Financial models rely on simulating many independent random paths. The CLT guarantees that simulation averages converge to the true expected value, making simulation results reliable.

f) Central banking and policy. Central banks use the CLT when analyzing aggregate economic data (inflation, GDP growth) composed of many individual transactions, allowing them to set policy based on sample estimates with known precision.

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5. Conclusion

The CLT states that the standardized sum (or mean) of a large number of i.i.d. random variables with finite mean and variance converges to the standard normal distribution. Its derivation via moment generating functions shows that normality emerges naturally from aggregation. In finance, this theorem underpins portfolio theory, risk management (VaR), option pricing, sampling inference, insurance pooling, and simulation — making it one of the most practically significant results in financial mathematics.

Time Series Data — Definition, Examples, and Information in Financial Time Series

Definition

Time series data is a sequence of observations of a variable recorded at successive points in time, usually at equal intervals. In symbols, if a variable $X$ is observed at times $t_1, t_2, \dots, t_n$, the resulting series

$X_{t_1}, X_{t_2}, \dots, X_{t_n} \quad \text{or simply} \quad \{X_t\}$

is called a time series. The essential feature is that the observations are ordered in time, so that past values influence or are related to present and future values.

Examples

1. Economic: Yearly GDP of Nepal from 2000 to 2024; monthly inflation rates.
2. Financial: Daily closing price of the NEPSE index; monthly interest rates; quarterly dividend payments of a commercial bank.
3. Physical/Natural: Daily temperature of Kathmandu; annual rainfall recorded at a station.
4. Business: Monthly sales of a departmental store; weekly demand for a product.

Information Contained in a Financial Time Series

A financial time series (e.g., stock prices, exchange rates, interest rates) typically contains the following information:

1. Trend: The long-run upward or downward movement of the series over time (e.g., a steadily rising share price).
2. Seasonal variation: Regular periodic fluctuations repeating within a year (e.g., higher bank deposits during festive seasons).
3. Cyclical variation: Long-term wave-like movements of more than one year, often linked to business cycles (e.g., boom and recession phases in stock markets).
4. Irregular (random) component: Unpredictable, erratic fluctuations caused by sudden events such as political instability, natural disasters, or policy changes.
5. Volatility: The degree of variation of returns over time; financial series often show volatility clustering (large changes followed by large changes).
6. Autocorrelation and dependence structure: The correlation of a series with its own past values, which is the basis of forecasting models such as AR, MA, and ARIMA.
7. Distributional properties: Mean, variance, skewness, and kurtosis of returns — financial returns often exhibit fat tails and leptokurtosis.
8. Stationarity/non-stationarity: Whether the statistical properties remain constant over time; price series are usually non-stationary while return series are often stationary.

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OR

Show that for a Stationary Stochastic Process: $C(\tau) \sim R(\tau)$

Setup and Definitions

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

Autocovariance function:

$C(\tau) = \text{Cov}(X_t, X_{t+\tau}) = E\big[(X_t - \mu)(X_{t+\tau} - \mu)\big]$

Autocorrelation function:

$R(\tau) = \text{Corr}(X_t, X_{t+\tau}) = \frac{\text{Cov}(X_t, X_{t+\tau})}{\sqrt{\text{Var}(X_t)\,\text{Var}(X_{t+\tau})}}$

Derivation

Since the process is stationary, $\text{Var}(X_t) = \text{Var}(X_{t+\tau}) = \sigma^2$ for all $t$. Substituting into the definition of $R(\tau)$:

$R(\tau) = \frac{C(\tau)}{\sqrt{\sigma^2 \cdot \sigma^2}} = \frac{C(\tau)}{\sigma^2}$

Hence,

$\boxed{R(\tau) = \frac{C(\tau)}{\sigma^2} \quad \text{and equivalently} \quad C(\tau) = \sigma^2 R(\tau)}$

Conclusion

Since $\sigma^2$ is a constant for a stationary process, the autocovariance $C(\tau)$ and the autocorrelation $R(\tau)$ differ only by the constant factor $\sigma^2$. Therefore,

$C(\tau) \sim R(\tau)$

i.e., they are proportional to each other. In particular, at lag $\tau = 0$:

$R(0) = \frac{C(0)}{\sigma^2} = \frac{\sigma^2}{\sigma^2} = 1$

which confirms that the autocorrelation at zero lag is always unity. Thus for a stationary stochastic process, the autocovariance and autocorrelation functions carry the same information about the dependence structure of the series, differing only by a scale factor.

a) Chaos approach in market

The chaos approach in finance holds that financial markets are complex, non-linear, dynamic systems whose price movements, though deterministic in principle, are so sensitive to initial conditions that they appear random and are practically unpredictable. It rejects the efficient market hypothesis view that prices move as a random walk around equilibrium.

Key features:

• Non-linearity: Prices do not respond proportionally to changes in fundamentals; small causes can produce large effects.
• Sensitive dependence on initial conditions (the "butterfly effect"): A tiny change in today's price or expectation can produce a completely different price path later. Long-run prediction is therefore impossible.
• Deterministic chaos: The system is governed by fixed rules (no true randomness), yet the resulting time series looks irregular and noisy.
• Strange attractors and fractals: Price series tend to settle into a bounded but never-repeating pattern (a strange attractor), and they show self-similar (fractal) structure across time scales.
• Bounded instability: Prices fluctuate within a range but never settle to a fixed point or a simple cycle.

Implication for finance: markets are inherently unstable and disequilibrium-prone; technical patterns may have limited predictive value; risk cannot be fully captured by normal-distribution models; and diversification and dynamic hedging matter more than static equilibrium models. Tools such as phase-space reconstruction, Lyapunov exponents and Hurst exponents are used to test for chaos in price data.

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b) Martingales

A martingale is a stochastic process $\{X_t\}$ such that the expected value of the next observation, given all past information, equals the present value:

$E[X_{t+1} \mid X_t, X_{t-1}, \dots, X_1] = X_t$

In words, the best forecast of tomorrow's value is today's value; the process has no drift and no predictable gain. A process is a submartingale if $E[X_{t+1}\mid \mathcal{F}_t] \ge X_t$ (favourable game) and a supermartingale if $E[X_{t+1}\mid \mathcal{F}_t] \le X_t$ (unfavourable game).

Example — fair coin-tossing game: A player starts with Rs 0 and wins Rs 1 for heads, loses Rs 1 for tails. Let $X_n$ be the cumulative gain after $n$ tosses. Then

$E[X_{n+1}\mid X_n] = X_n + \tfrac12(+1) + \tfrac12(-1) = X_n$

so $\{X_n\}$ is a martingale. The player's expected wealth at any future time equals his present wealth — a "fair game".

Example — stock prices: If a stock price $S_t$ follows a martingale, then $E[S_{t+1}\mid S_t] = S_t$, meaning the expected return over one period is zero. Under the risk-neutral measure, discounted stock prices are martingales, which is the basis of option pricing (Black–Scholes).

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c) Markov and non-Markov processes

Markov process: A stochastic process in which the future depends only on the present state, not on the past history. It satisfies the Markov property:

$P(X_{t+1} = j \mid X_t = i, X_{t-1}, \dots, X_0) = P(X_{t+1} = j \mid X_t = i)$

The present state contains all information needed to predict the future. If the state space is discrete, it is a Markov chain, described by a transition probability matrix $P = [p_{ij}]$ where $p_{ij} = P(X_{t+1}=j \mid X_t=i)$ and $\sum_j p_{ij} = 1$.

Example: A simple weather model — if it rains today, the chance it rains tomorrow is 0.7; if it is dry today, the chance of rain tomorrow is 0.3. Tomorrow's weather depends only on today's, not on the weather of previous days. Random walks and Poisson processes are also Markov.

Non-Markov process: A process in which the future depends on the past beyond the present state — i.e., the Markov property fails. History matters.

Example: A stock price with momentum — if prices have been rising for several days, the probability of a further rise may be higher than if they had been falling. Here the whole past trend, not just today's price, influences tomorrow. Similarly, a person's health status may depend on the entire history of illnesses, not only on the current state. Such processes require models with memory, e.g. autoregressive (AR) models or higher-order Markov chains.

Key difference: Markov process = memoryless (present state sufficient); non-Markov process = has memory (past path matters).

Arbitrage

Arbitrage is the simultaneous purchase and sale of the same asset (or equivalent assets) in two different markets at prices that are temporarily out of line, in order to earn a riskless profit from the price difference. The arbitrageur buys in the cheaper market and sells in the dearer market at the same moment, so no investment is exposed to price risk and no net capital is committed.

Illustration

Suppose the same share of Nabil Bank is traded on two exchanges:

An arbitrageur buys 1,000 shares on Exchange A for Rs 500,000 and simultaneously sells 1,000 shares on Exchange B for Rs 520,000.

$\text{Profit} = (520 - 500) \times 1{,}000 = \text{Rs } 20{,}000$

(ignoring brokerage and transaction costs). The profit is locked in at the instant of the two trades — it does not depend on which way the price moves afterwards.

How arbitrage eliminates its own opportunity

The arbitrageur's actions themselves change prices:

1. Buying pressure in the cheap market — the extra demand on Exchange A pushes its price up from Rs 500.
2. Selling pressure in the expensive market — the extra supply on Exchange B pushes its price down from Rs 520.
3. This process continues as long as a price gap exists. It stops only when the two prices become equal (or differ by no more than transaction costs), i.e. when the law of one price is restored.

At that point the arbitrage profit is zero and the opportunity disappears. Because arbitrage is visible to many traders, competition makes this adjustment very fast — often within minutes in liquid markets — so mispricing is small and short-lived. This is why arbitrage is described as self-eliminating: the very act of exploiting a price difference removes it, and it is this mechanism that keeps market prices consistent and efficient.

The mean is $\mu=\overline{S}(n)=3.0$ and the variance is $\sigma^2=3$, so $\sigma=\sqrt3$.

The Gaussian (normal) probability density is

$P(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\!\left[-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right]$

Substituting $\mu=3$ and $\sigma^2=3$:

$P(x)=\frac{1}{\sqrt{6\pi}}\exp\!\left[-\frac{(x-3)^{2}}{6}\right],\qquad -\infty<x<\infty$

Numerically $\frac{1}{\sqrt{6\pi}}\approx 0.230$, so the peak value at $x=3$ is $P(3)\approx 0.23$.

Check: the density is symmetric about $x=3$, it is normalised ($\int_{-\infty}^{\infty}P(x)\,dx=1$), and it has $\langle x\rangle=3$ and $\langle (x-3)^2\rangle=3$ as required. About 68 % of the probability lies within $3\pm\sqrt3$, that is between $1.27$ and $4.73$.

Let (X\sim \operatorname{Gamma}(\gamma,1)), so its probability density function is

[ f_X(x)=\frac{x^{\gamma-1}e^{-x}}{\Gamma(\gamma)},\qquad x>0,\ \gamma>0. ]

1. Characteristic function of the Gamma distribution

The characteristic function of (X) is

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

Substitute the density:

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

Let

[ u=(1-iq)x. ]

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

[ dx=\frac{du}{1-iq},\qquad x^{\gamma-1}=\frac{u^{\gamma-1}}{(1-iq)^{\gamma-1}}. ]

Hence

[ \phi_X(q)=\frac{1}{\Gamma(\gamma)(1-iq)^{\gamma}}\int_0^\infty u^{\gamma-1}e^{-u},du. ]

But

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

Therefore,

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

Thus,

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

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2. Infinite divisibility of the Gamma distribution

Yes, the Gamma distribution is infinitely divisible.

A distribution is infinitely divisible if for every positive integer (n), the random variable can be expressed as the sum of (n) independent and identically distributed random variables.

Let

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

and suppose (Y_{n,1},Y_{n,2},\dots,Y_{n,n}) are independent. The characteristic function of each (Y_{n,k}) is

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

Define

[ S_n=Y_{n,1}+Y_{n,2}+\cdots+Y_{n,n}. ]

Since the variables are independent, the characteristic function of (S_n) is the product of their characteristic functions:

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

But this is exactly the characteristic function of (X\sim \operatorname{Gamma}(\gamma,1)):

[ \phi_X(q)=(1-iq)^{-\gamma}. ]

By the uniqueness theorem for characteristic functions,

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

Therefore, for every (n),

[ X \overset{d}{=} Y_{n,1}+Y_{n,2}+\cdots+Y_{n,n}, ]

where the (Y_{n,k}) are iid (\operatorname{Gamma}(\gamma/n,1)) variables. Hence the Gamma distribution is infinitely divisible.

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

Moreover, this infinite divisibility allows the construction of a Gamma process, a Lévy process whose increments have Gamma distributions.

Option 1 — Physics in socio-economic problems

The basic idea

A socio-economic system — a market, a city, a population of voters — is a complex system: very many interacting agents, each following simple rules, producing collective behaviour that nobody designed. Physics has spent a century learning how to handle exactly this kind of problem (many particles, short-range interactions, emergent collective states). So physicists borrow its tools — order parameters, critical points, feedback, randomness, scaling laws — and apply them to economics and society. The field is called econophysics (or sociophysics).

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1. Phase transitions

Concept used: a system changes qualitatively when a control parameter (temperature, noise, connectivity) crosses a critical value. Near the critical point the order parameter goes to zero, fluctuations become enormous, and the correlation length diverges as a power law.

Examples:

• Opinion formation (voter / Ising model). Let each person hold opinion $+1$ or $-1$. Neighbours copy each other (ferromagnetic coupling), while "noise" (independent thinking, media influence) acts like temperature $T$. The order parameter is the average opinion $m=\frac1N\sum_i s_i$. Below a critical noise $T_c$ society is consensus (one dominant opinion); above $T_c$ it is fragmented into competing groups. This explains why a small rise in noise or a change in communication structure can suddenly destroy a national consensus.
• Schelling's segregation model. Agents mildly prefer neighbours "like themselves". Below a threshold preference the city stays mixed; just above it, the whole city abruptly separates into ghettos — a phase transition, not a gradual drift. This is a classic explanation of urban segregation.
• Financial crashes. Sornette's log-periodic power law model treats a bubble as a system approaching a critical point $t_c$: $\ln p(t)=A+B\,(t_c-t)^m\Big[1+C\cos\big(\omega\ln(t_c-t)-\phi\big)\Big]$ The accelerating oscillations before the crash are the "critical fluctuations" of a phase transition. The crash itself is the symmetry-breaking event.
• Technology / product adoption. Adoption spreads through a social network like a percolation process. If the fraction of early adopters exceeds a threshold, the innovation "percolates" and takes over the whole market; below it, it dies out. This is why marketing focuses on "critical mass".

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2. Non-linear dynamics

Concept used: when feedback is non-linear, small causes need not give small effects. The system can show limit cycles, bifurcations, chaos, and sensitive dependence on initial conditions (the "butterfly effect"), so long-term prediction is impossible even though the rules are deterministic.

Examples:

• Business cycles. Samuelson's multiplier–accelerator model and Goodwin's predator–prey model of the wage-profit struggle are non-linear oscillator models. Goodwin's equations $\frac{\dot u}{u}=\alpha-\beta v,\qquad \frac{\dot v}{v}=-\gamma+\delta u$ are exactly Lotka–Volterra: employment $v$ and wage share $u$ chase each other in endless cycles — booms and slumps without any external shock.
• Cobweb model of farm prices. Farmers plant according to last year's price: $Q_t^s=c+dP_{t-1}$, $Q_t^d=a-bP_t$. If supply responds more strongly than demand ($d>b$), the price sequence diverges or becomes chaotic — a mathematical reason for violent price swings in agricultural markets.
• Chaotic time series in finance. Stock returns have positive Lyapunov exponents and fractal structure; Mandelbrot showed price charts are statistically self-similar (fractal), so "trend lines" have limited predictive value.
• Traffic flow. Non-linear car-following models show a sharp transition from free flow to a jam at a critical density — a phase transition in a purely social system.

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3. Disordered systems

Concept used: real materials contain quenched randomness (frozen impurities). The prototype is the spin glass: random competing interactions create frustration — no single configuration satisfies all bonds — so the energy landscape has an astronomical number of nearly equal metastable states, separated by barriers. The system ages, remembers its history, and is non-ergodic.

Examples:

• Lock-in and path dependence in economics. A market with increasing returns (network effects) is a frustrated system: several technologies are locally optimal. Random early events decide which one "freezes in" — the QWERTY keyboard versus the Dvorak layout, VHS versus Betamax. History, not efficiency, selects the winner. This is spin-glass behaviour in an economy.
• Wealth and income distribution. Random multiplicative growth (Gibrat's law) gives a log-normal distribution; adding a lower bound or "reset" produces a Pareto power law $P(X\ge x)\propto x^{-\alpha}$ for the rich tail. Pareto's original observation that ~20 % of people hold ~80 % of wealth is a power law, and it arises from disordered multiplicative dynamics.
• Portfolio optimisation and neural networks. The Hopfield model of memory is mathematically a spin glass; the same replica methods are used to find optimal portfolios and to study the rugged "fitness landscapes" of firms and technologies.
• Random-field Ising model reproduces the observed avalanche (power-law) distribution of firms' size changes and of social-influence cascades.

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4. Several other areas

• Self-organized criticality. The Bak–Tang–Wiesenfeld sandpile model shows that a slowly driven system with thresholds drives itself to a critical state, producing avalanches of all sizes: $P(s)\sim s^{-\tau}$. This explains the power laws in earthquake magnitudes (Gutenberg–Richter), forest fires, solar flares, and stock-market crashes — big crashes are not anomalies but the natural tail of the avalanche distribution.
• Network theory. Social and economic networks are scale-free ($P(k)\sim k^{-\gamma}$, $2<\gamma<3$) and small-world. This explains the speed of epidemics and rumours, the "six degrees of separation", and why the failure of a few highly connected banks can collapse the whole financial system (2008 crisis).
• Statistical mechanics of games. Evolutionary game theory plus physics gives the emergence of cooperation, the emergence of money as a medium of exchange, and the dynamics of tax evasion and corruption.
• Entropy and information theory. Maximum-entropy models reproduce observed income distributions; the Theil index is literally an entropy measure of inequality. Shannon entropy measures the information content of a market or a language.
• Fractals and scaling. Mandelbrot's fractal geometry describes price series, city sizes (Zipf's law), and firm sizes.

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5. Conclusion

Physics contributes to socio-economics mainly through quantitative, testable models of collective behaviour: order parameters and critical points for sudden social change, non-linear dynamics for cycles and chaos, disorder theory for lock-in and inequality, and scaling laws for avalanches and networks. Its limits must be stated honestly: unlike atoms, people learn, anticipate and react (reflexivity), there are no conservation laws in economics, and data are noisy and non-repeatable. Physics analogies are therefore powerful tools, not proofs — they must be tested against real socio-economic data before being used for policy.

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OR — The St. Petersburg Paradox and its power-law formulation

Statement of the paradox

The paradox was posed by Nicolas Bernoulli (1713) and published by Daniel Bernoulli (1738).

A fair coin is tossed until the first head appears. If the head appears on the $n$-th toss, the player receives a prize of $2^n$ rupees. The question: what is a fair price to pay for one play?

The probability that the game ends on the $n$-th toss is $P(N=n)=\left(\frac12\right)^{n}$. The expected payoff is therefore

$E(X)=\sum_{n=1}^{\infty}2^{-n}\cdot 2^{n}=\sum_{n=1}^{\infty}1=1+1+1+\cdots=\infty$

The expected value is infinite, so expected-value theory says a rational person should pay any finite sum to play. Yet in practice nobody will pay more than a few rupees. This contradiction between the theory of expected value and actual human behaviour is the St. Petersburg Paradox.

Classical resolutions

• Daniel Bernoulli's expected utility. Utility is not money but a concave function of money, $u(x)=\ln x$. Then $E[u]=\sum_{n=1}^{\infty}2^{-n}\ln 2^{n}=\ln 2\sum_{n=1}^{\infty}\frac{n}{2^{n}}=2\ln 2=\ln 4$ so the certainty equivalent is $e^{\ln 4}=\text{Rs }4$. A rational player pays about Rs 4 — matching intuition. (Cramer had earlier proposed $u=\sqrt{x}$.)
• Buffon's empirical argument: in practice the casino's capital is finite, so the game cannot run beyond a certain $n$.
• Modern view: the paradox is really about fat tails, not about utility.

Formulation in terms of a power law

Write the payoff as a continuous variable $x=2^{n}$, so $n=\log_2 x$. The survival (complementary) distribution of the payoff is

$P(X\ge x)=P(N\ge n)=\sum_{k=n}^{\infty}2^{-k}=2^{-(n-1)}=\frac{2}{2^{\,n}}=\frac{2}{x}$

Hence

$\boxed{P(X\ge x)\;\propto\;x^{-\alpha},\qquad \alpha=1}$

This is exactly a power law (Pareto) distribution with exponent $\alpha=1$. Equivalently, differentiating the survival function gives the density

$f(x)=-\frac{d}{dx}P(X\ge x)=\frac{2}{x^{2}},\qquad x\ge 2,$

which is a power law $f(x)\propto x^{-(\alpha+1)}=x^{-2}$.

Why the paradox appears

For a power-law density $f(x)\propto x^{-(\alpha+1)}$ the mean is

$\langle X\rangle=\int_{x_{\min}}^{\infty}x\,f(x)\,dx\;\propto\;\int^{\infty}x^{-\alpha}\,dx,$

which converges only if $\alpha>1$. Here $\alpha=1$ exactly, so the integral diverges logarithmically and the mean is infinite. The variance (needing $\alpha>2$) is also infinite.

Because the tail is so heavy, the sample average of $N$ plays does not settle down: it grows roughly like $\log_2 N$, and is dominated by rare, huge payoffs. Typical players therefore see small winnings, and the infinite "average" is meaningless for any finite number of plays.

Conclusion

The St. Petersburg paradox is a heavy-tailed (power-law) distribution with exponent $\alpha=1$, $P(X\ge x)=2/x$ and $f(x)=2/x^2$. Its infinite mean explains why expected value is a poor guide here, and why people (and economists, through utility or finite-bankroll arguments) value the game at only a few rupees.

Random Walk: Expectation and Variance of the Sum $S_n$

1. Setup

Let $X_1, X_2, \dots, X_n$ be independent and identically distributed (i.i.d.) random variables. The random walk is defined by the partial 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

$E(X_i) = \mu, \qquad \operatorname{Var}(X_i) = \sigma^2 \quad \text{for all } i = 1, 2, \dots, n$

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2. Expectation of $S_n$

Using the linearity of expectation (which holds whether or not the variables are independent):

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

Since each $X_i$ has the same mean $\mu$,

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

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

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

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3. Variance of $S_n$

By definition,

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

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

Expanding the square,

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

The first term gives $\sum_{i=1}^{n} \sigma^2 = n\sigma^2$.

For the cross terms, since $X_i$ and $X_j$ are independent for $i \neq j$,

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

Hence all covariance terms vanish, and

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

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

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4. Standard Deviation

$SD(S_n) = \sqrt{\operatorname{Var}(S_n)} = \sigma\sqrt{n}$

This shows the √n law: the spread of the walk grows with the square root of the number of steps.

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5. Special Case: Simple Symmetric Random Walk

If each step is $+1$ or $-1$ with probability $\tfrac12$ each:

$E(X_i) = (+1)\tfrac12 + (-1)\tfrac12 = 0 \;\Rightarrow\; E(S_n) = 0$

$E(X_i^2) = (+1)^2\tfrac12 + (-1)^2\tfrac12 = 1 \;\Rightarrow\; \operatorname{Var}(X_i) = 1 - 0^2 = 1$

$\therefore \operatorname{Var}(S_n) = n \cdot 1 = n, \qquad SD(S_n) = \sqrt{n}$

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6. Conclusion

For $n$ i.i.d. random variables forming the random walk $S_n = \sum_{i=1}^{n} X_i$:

The key reasons are linearity of expectation for the mean and the independence of the variables (which makes all covariance terms zero) for the variance.

Stochastic Process

A stochastic process is a collection of random variables $\{X_t\}$ indexed by a parameter $t$ (usually time), defined on a common probability space. Formally, for a given index set $T$ (often $T = \{0,1,2,\dots\}$ or $T = [0,\infty)$), a stochastic process is a family

$\{X_t : t \in T\}$

where each $X_t$ is a random variable. In other words, it is a mathematical model of a system that evolves over time in a way governed by chance — the value at any future time cannot be predicted with certainty, only described probabilistically.

Key features:

• State space: the set of all possible values the process can take.
• Index set (parameter set): the set of times at which the process is observed.
• Discrete vs continuous: if both the index set and state space are discrete, the process is a discrete-time, discrete-state process (e.g., a random walk); if time is continuous, it is a continuous-time process (e.g., Brownian motion).

Examples: daily closing prices of a stock, the number of customers arriving at a bank per hour (Poisson process), the movement of a particle in a fluid (Brownian motion), and exchange rates over time.

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Stationary Stochastic Process

A stochastic process is said to be stationary if its statistical properties do not change over time — that is, the behaviour of the process is invariant with respect to a shift in the time origin.

Strict (Strong) Stationarity

A process $\{X_t\}$ is strictly stationary if, for any set of times $t_1, t_2, \dots, t_n$ and any shift $k$, the joint probability distribution of

$(X_{t_1}, X_{t_2}, \dots, X_{t_n})$

is identical to that of

$(X_{t_1+k}, X_{t_2+k}, \dots, X_{t_n+k}).$

That is, the whole distribution depends only on the relative spacing of the times, not on the actual calendar time.

Weak (Second-order) Stationarity

A process is weakly stationary if it satisfies three conditions:

1. Constant mean: $E(X_t) = \mu$ for all $t$.
2. Constant variance: $\operatorname{Var}(X_t) = \sigma^2 < \infty$ for all $t$.
3. Autocovariance depends only on the lag: $\operatorname{Cov}(X_t, X_{t+k}) = \gamma(k)$, a function of the time gap $k$ alone, not of $t$.

Properties and Significance

• Strict stationarity implies weak stationarity (provided the first two moments exist); the converse is not generally true, except for the normal distribution, where the two coincide.
• Stationarity is the basic requirement for many time-series models (AR, MA, ARMA) and for forecasting, since a non-stationary series must first be differenced or transformed to make it stationary.
• A white noise process is the simplest example of a stationary process: constant zero mean, constant variance, and zero autocovariance at all non-zero lags.

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OR

Time and Price Scales in Financial Markets

Time Scale

The time scale refers to the frequency and interval at which financial data are recorded and analysed. It determines the resolution of observation of price movements.

• Tick-by-tick (high frequency): every single transaction is recorded; used in algorithmic and high-frequency trading.
• Intraday: data recorded at intervals within a trading day — e.g., 1-minute, 5-minute, 15-minute, hourly bars.
• Daily: one observation per trading day (open, high, low, close); the most common scale for technical and fundamental analysis.
• Weekly, monthly, quarterly, yearly: longer horizons used for medium- and long-term trend analysis and portfolio decisions.

The choice of time scale affects volatility estimates, the number of observations, and the statistical properties of returns. Returns computed over different time scales are related by the square-root-of-time rule for scaling volatility:

$\sigma_{\text{annual}} = \sigma_{\text{daily}} \times \sqrt{252}$

(assuming 252 trading days per year).

Price Scale

The price scale refers to the way prices are represented and plotted on the vertical axis of a chart. The two common types are:

• Arithmetic (linear) scale: equal vertical distances represent equal absolute price changes. A move from Rs 10 to Rs 20 looks the same as a move from Rs 100 to Rs 110. Suitable for short-term charts and small price ranges.
• Logarithmic (log) scale: equal vertical distances represent equal percentage changes. A move from Rs 10 to Rs 20 (100% rise) looks the same as a move from Rs 100 to Rs 200. Preferred for long-term charts and for assets with wide price ranges, since it shows proportional changes consistently.

Importance

• The time scale governs the data frequency and sample size, influencing volatility, correlation, and risk measurement.
• The price scale governs the visual interpretation of trends and patterns; using a log scale avoids distortion when prices move over wide ranges.
• Together they affect chart analysis, model selection, and trading strategy, so analysts must choose scales consistent with their investment horizon and objectives.

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 increases. It provides a mathematical framework for comparing algorithms independently of the machine on which they run.

Key points:

• Input size (n): the amount of data the algorithm processes.
• Time complexity: number of basic operations as a function of n.
• Space complexity: amount of extra memory used as a function of n.
• Asymptotic notation: Big-O gives an upper bound, Big-Omega a lower bound, and Big-Theta a tight bound on growth.
• Common classes: O(1) constant, O(log n) logarithmic, O(n) linear, O(n log n), O(n²) quadratic, O(2ⁿ) exponential.
• Tractable vs intractable: problems solvable in polynomial time are tractable; those requiring exponential time are intractable.
• Complexity classes: P (solvable in polynomial time), NP (verifiable in polynomial time), NP-complete (hardest problems in NP), and NP-hard.
• Importance: helps in choosing efficient algorithms, predicting performance on large inputs, and understanding the limits of computation.

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b) Meaning of volatility

Volatility is a statistical measure of the dispersion or variability of returns of an asset (such as a stock, index, or currency) around its average value. It indicates how much and how quickly the price of an asset changes over time.

Key points:

• High volatility: prices fluctuate widely and rapidly; the asset is risky but offers higher potential returns.
• Low volatility: prices move slowly and steadily; the asset is relatively stable and less risky.
• Measurement: usually measured by the standard deviation or variance of returns; in finance, annualised standard deviation is common.
• Types: historical (backward-looking) volatility and implied (forward-looking) volatility derived from option prices.
• Uses: risk assessment, portfolio management, option pricing (e.g., Black–Scholes model), and value-at-risk estimation.
• Interpretation: volatility is often called "risk" in finance, though it measures both upside and downside movement.

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c) White noise with suitable example

Definition: White noise is a random sequence of variables that has a constant mean (usually zero), constant variance, and zero autocorrelation at all lags. In other words, its values are uncorrelated over time and show no predictable pattern.

Formally, a series $\varepsilon_t$ is white noise if:

$E(\varepsilon_t) = 0, \quad Var(\varepsilon_t) = \sigma^2, \quad Cov(\varepsilon_t, \varepsilon_{t-k}) = 0 \text{ for } k \neq 0$

Example: Tossing a fair coin repeatedly and recording +1 for heads and −1 for tails gives a white noise series. Each toss is independent of the previous ones, the mean is zero, the variance is constant, and there is no correlation between successive outcomes. Similarly, the residuals of a well-fitted regression or ARIMA model should behave like white noise.

"The market doesn't have memory" — justification and explanation

Meaning of the statement

The statement means that past prices do not influence or determine future prices. Whatever happened to the price yesterday, last month or last year is already fully reflected in today's price. The market only reacts to new, unexpected information as it arrives. Hence the market is said to be "memoryless" — it does not "remember" its past.

Theoretical justification (Efficient Market Hypothesis)

According to the Efficient Market Hypothesis (EMH) in its weak form:

• All historical price and volume information is already incorporated into the current market price.
• Prices follow a random walk — the next price change is independent of all previous price changes.
• Only new information (news, earnings, policy changes) can move the price, and since news is unpredictable, price changes are unpredictable.

Therefore, studying past price patterns gives no reliable advantage — the market has no memory.

Explanation with an example (graph)

Suppose the price of a stock over five days is:

Plotting price on the vertical axis against time on the horizontal axis gives a zig-zag line:

Price (Rs)
110 |        *
105 |              *
100 |  *                 *
 95 |           *
    +----------------------------> Time
       D1   D2   D3   D4   D5

Interpretation:

• The price rises from Rs 100 to Rs 110, falls to Rs 95, rises to Rs 105, and falls back to Rs 100.
• Each movement was caused by new information arriving on that day, not by the previous day's price.
• The path shows no reliable pattern — knowing that the price rose on Day 2 does not tell us it will rise on Day 3 (in fact it fell).
• This random, patternless movement confirms that the market does not remember its past; it reacts only to fresh news.

Conclusion

Since current prices already contain all past information and future changes depend only on unpredictable new information, past price behaviour cannot be used to predict future prices. This is why it is said that the market doesn't have memory.

For a one-dimensional random walk, the diffusion constant is

[ D=\frac{(\text{step size})^2}{2(\text{time step})} ]

Given:

[ \text{Step size } a=1.5\ \text{ft},\qquad \text{Time step } \tau=0.9\ \text{s} ]

Substituting:

[ D=\frac{(1.5)^2}{2(0.9)} ]

[ D=\frac{2.25}{1.8}=1.25\ \text{ft}^2/\text{s} ]

In SI units, using (1\ \text{ft}=0.3048\ \text{m}),

[ D=1.25(0.3048)^2 ]

[ D=0.116\ \text{m}^2/\text{s} ]

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

If the random walk is taken as three-dimensional, the relation is (D=a^2/(6\tau)), giving

[ D=\frac{(1.5)^2}{6(0.9)}=0.4167\ \text{ft}^2/\text{s}=0.0387\ \text{m}^2/\text{s} ]

The usual one-dimensional result is (\boxed{1.25\ \text{ft}^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 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 $X$ is infinitely divisible if, for every positive integer $n$, there exist $n$ 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 satisfies

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

where $\phi_n(q)$ is itself a valid characteristic function.

Take $X_i \sim \text{Poisson}\!\left(\dfrac{\lambda}{n}\right)$, $i = 1, 2, \dots, n$, all independent. The characteristic function of each $X_i$ is, by the result above,

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

Since the sum of independent Poisson variates is again Poisson with parameter equal to the sum of the parameters,

$X_1 + X_2 + \cdots + X_n \sim \text{Poisson}\!\left(\underbrace{\frac{\lambda}{n} + \frac{\lambda}{n} + \cdots + \frac{\lambda}{n}}_{n \text{ terms}}\right) = \text{Poisson}(\lambda).$

Therefore

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

Since $\phi_n(q)$ is the characteristic function of the Poisson$\left(\lambda/n\right)$ distribution (a genuine characteristic function) for every $n$, the Poisson characteristic function is an $n$-th power of a characteristic function for every $n$. Hence the Poisson distribution is infinitely divisible, and the corresponding Poisson process is an infinitely divisible random process.